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Geometric and Topological Structures Related to Universal Algebras
Research Institute for Systems Research, Russian Academy of Sciences, 109280 Moscow, Russia

M. I. Graev and A. V. Koganov
Received December 23, 2002

Abstract.

system is closed with respect to the operations and forms a semimodular lattice similar to the system of subspaces of a pro jective space. This structure enables one to extend the metric and topology from the system of generators of the A-system to the entire A-system in such a way that the Archimedean and Hausdor properties are preserved under the extension.

A-systems) that includes free algebras with an arbitrary set of operations, their commutative and idempotent modi cations, and some other ob jects. It turns out that there is a system of subsets (the so-called planar subsets) of the support of an arbitrary tame A-system, and this

In the paper, a special class of universal algebras is introduced (the so-called tame

INTRODUCTION 1. The paper is devoted to universal algebras. Recall that a universal algebra is de ned by a set U with an arbitrarily given collection F of mappings f : U n ! U , n = 0 1 2 : :: where the number n depends n f 2 F . The set U is referred to as the support of the universal algebra and on the mappings f : U ! U as n-ary operations on U , see 1, 2]. The universal algebra with support U and the set of operations F is denoted in the paper by (U F ), the subset of n-ary operations of U by Fn , and the image of a sequence (u1 ::: un ) 2 U n under an n-ary operation f by f (u1 ::: un ). It is assumed that F contains no 0-ary operations ? ! U (under a 0-ary operation, some element u 2 U becomes xed). For brevity, instead of the term \universal algebra" we always use the term \A-system." The theory of universal algebras has been developing extensively since the forties of the last century. Since the notion of universal algebra is sometimes too general, the investigations in this area are related to some restrictions concerning the operations in F . 2. In the paper, we consider A-systems (universal algebras) (U F ) satisfying the following two conditions. Condition 1 (uniqueness of decompositions). For any element u 2 U , there exists at most one representation (up to the order of elements u1 ::: un ) of the form u = f (u1 ::: un) f 2F (1) where ui 6= u for at least one index i. Condition 2. The support U is generated by the subset of elements in U which are not representable in the form (1). This means that every element u 2 U can be obtained from these elements by a composition of nitely many operations f 2 F . We refer to the universal algebras of this kind as tame A-systems. The class of tame A-systems includes all free, free commutative, and free idempotent A-systems. An A-system (U F )is saidto be free if (1) the relation f (u1 ::: um )= f 0 (u01 ::: u0n ) (2) holds if and only if f = f 0 (and hence m = n)and ui = u0i for each i (2) there are no relations of the form u = f (u : : : u), and (3) the support U is generated by the subset of elements in U not representable in the form u = f (u1 ::: un).
Supported by RFBR under grant no. 01-01-00754.

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The simplest examples of free A-systems are as follows: 1) a sequence fx1 ::: xn ::: g with the only unary operation f (xn ) = xn+1 2) U is the set of \words" in the one-letter alphabet: a aa a(aa) (aa)a (aa)(aa) etc. the only binary operation given on U is the concatenation f (u v )= uv (juxtaposition of words). An A-system (U F ) is said to be free commutative if Condition 1 of the previous de nition is replaced by a weaker one, namely, relation (2) holds if and only if f = f 0 and the sequences (u1 ::: um ) and (u01 ::: u0n ) di er in their order only. An A-system (U F ) is said to be free idempotent if u = f (u ::: u) for all u 2 U and f 2 F and there are no other relations. 3. The tame A-systems (U F ) are endowed with several important structures. These are, rst, integral-valued characteristics of elements of the support, namely, the height of an element (the number of successive iterations of operations used to construct the given element from the indecomposable ones) and the length of an element u (the number of symbols of elements in a \word" expressing u in terms of indecomposable elements). We stress that, in a tame A-system, the length and the height of an element are de ned uniquely. Further, to any element u of a tame A-system one can assign a nite directed graph of tree type (decomposition scheme) whose vertices are operations used when constructing this element from indecomposable elements. The support U of anytame A-system is naturally endowed with a partial order relation. Finally,to any tame A-system, one can assign a directed graph whose vertices are elements of the support of this system. For any free A-system, this graph completely reconstructs the A-system up to isomorphism. These structures are studied in Section 2. 4. In Section 3 we assign to any tame A-system (U F ) an analog of pro jective space. We consider the family L of subsets V U that are closed with respect to the operations in F and have the nite base property. The number of base elements is called the rank of the set V and is denoted by r(V ). Note that an inclusion V1 V does not imply the inequality r(V1) 6 r(V ) because the set V can contain subsets of arbitrarily large rank. The set L is endowed with the structure of lattice with respect to the operation of intersection. However, this lattice is not semimodular, and therefore cannot be viewed as a geometric ob ject 4, 5]. In the lattice L,wechoose a family L0 L of elements V 2 L satisfying the following maximality condition : there exist no subsets V 0 2 L strictly containing V and such that r(V 0 ) 6 r(V ) these subsets are said to be planar. We prove that the family L0 of planar subsets is closed with respect to the operation of intersection, and therefore the sets L0 and L are simultaneously endowed with the structure of a lattice with respect to this operation. Note that the operations of union in L0 and L are di erent, and hence L0 is not a sublattice of L. It turns out that the lattice L0 is semimodular. Thus, one can naturally interpret the elements of V of rank r as planes of dimension r ; 1 in the \pro jective" (generally in nite-dimensional) space U because these ob jects satisfy the main axioms on the intersections and unions of planes. The speci c feature of the geometry thus de ned is that, on every plane of dimension n,wehave a uniquely chosen set of n + 1 points generating the plane (elements of the base). Here the base of the intersection of two planes is contained in the union of the bases of these planes. Starting from the notion of a planar subset, we then introduce the notion of plane in the support U of a tame A-system. An F -subset V U of a tame A-system A = (U F ) is called a plane if every planar subset in V is a planar subset in U . In particular, all planar subsets in U are planes. We prove that eachintersection of planes is also a plane. Thus, the set of planes is endowed with the structure of a lattice, and the family of planar subsets forms a sublattice of this lattice. It is shown that the lattice of planes is also semimodular. 5. Sections 4 and 5 are devoted to topological and metric universal A-systems. An A-system (U F ) is said to be topological (metric ) if the support U of this system is endowed with the structure of a topological (metric) space with respect to which all operations f 2 F are continuous.
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If (U F ) is a tame A-system, then, starting from a topology (metric) on U , one can de ne the topology (metric) on the set L of all subsets V U of nite rank that are closed with respect to the operations in F . Here we restrict ourselves to free, free commutative, and free idempotent A-systems only. We construct an extension of the Hausdor topology and metric, Archimedean and non-Archimedean, which are initially given on the base X U only, to the Hausdor topology (Archimedean and non-Archimedean metric, respectively) on the entire support U . In the case of free idempotent A-system, we construct an extension of the topology to X to another topology on U , whichis weaker than the previous one. In this new (\secondary") topology, bases of neighborhoods are subsets of U closed with respect to the operations in F and generated by sets open in the original topology. Similarly, a non-Archimedean metric on X can be extended to a new non-Archimedean metric on U . In this new metric, every ball is a subset of U generated by a ball in the original metric and closed with respect to the operations in F . On the base of the topology and metric constructed on U , we then construct a topology and metric on the family L of all nite-rank subsets V U closed with respect to the operations in F . It is proved that, with respect to the topology thus constructed, the family L0 of planar subsets paying the role of planes in our geometry forms an open dense set. 1. A-SYSTEMS 1:1: De nition of A-System By an A-system we mean a pair A = (U F ) consisting of a set U of elements (the support of the A-system) and a set F of operations f : U n ! U (the fundamental set). For any operation f , the parameter n = n(f ) can be an arbitrary nonnegativeinteger, which is called the arity of the operation f , and f itself is called an n-ary operation. The subset of all n-ary operations is denoted by Fn . By de nition, each 0-ary operation xes some element of the set U . The image of an arbitrary sequence (u1 ::: un ) 2 U n under an n-ary operation f is denoted by f (u1 ::: un ). Example. 1) A set U with a chosen set of mappings f : U ! U . 2) A set U with a single binary operation (multiplication) f : U U ! U (a groupoid) 3]. A homomorphism of one A-system into another and an isomorphism of two A-systems are de ned in the usual way. The de nition of A-system coincides with the traditional de nition of a universal algebra 1, 2], and, if additional relations are de ned on the elements of the support, with the de nition of an algebraic system. However, the last term was recently overloaded by other interpretations in various areas of mathematics. Since, in the present paper, we foresee introducing some additional relations, we use a special term \A-system" to avoid varying readings. In this paper, we assume that all A-systems under consideration contain no 0-ary operations. 1:2:F -Subsets and A-Subsystems Let A = (U F ) be an arbitrary A-system. Any subset U 0 U closed with respect to the operations f 2 F is said to be F -closed or, brie y, an F -subset. By de nition, an empty set is assumed to be F -closed. The A-systems of the form A0 = (U 0 F ), where U 0 U is an arbitrary F -subset, are said to be A-subsystems (or simply subsystems ) of the original A-system A =(U F ). Wesay that a subsystem (U 0 F )is embedded in a subsystem (U 00 F )if U 0 U 00 . If all subsets U U are F -subsets, then their intersection \U is also an F -subset, and the subsystem (\U F ) is called the intersection of the subsystems (U F ). Hence, the family of all F -subsets U 0 U and the set of subsystems A0 =(U 0 F )ofevery A-system A =(U F ) are endowed with the structures of lattices with respect to embedding. In these lattices, the products (the compositions) U1 ^ U2 and A1 ^ A2 of F -sets U1 and U2 and subsystems A1 =(U1 F )and A2 =(U2 F ) are the set U1 \ U2 and the subsystem (U1 \ U2 F ), respectively, and the sums (disjunctions) U1 _ U2 and A1 _ A2 are the intersection U 0 of all F -subsets containing U1 and U2 and the subsystem A0 =(U 0 F ), respectively.
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1:3: Generating and Basis Subsets For any A-system A =(U F ), denote by f (U 0 ), where U 0 U and f 2 F , the image of the set U n , n = n(f ), under the mapping U n ! U . Toany subset X U we assign the sequence of subsets X1 ::: Xn ::: (3) where X1 = X , and the sets Xn are de ned by induction on n,

Xn =
Obviously, their union

f 2F

f ((X

1

Xn;1 )
1
n
=1

n(f

)

):

U0 =

X

n

is an F -subset, and therefore the pair A0 =(U 0 F )isan A-subsystem of A. This subsystem is said to be generated by the subset X U .We then write U 0 = U (X ) A0 = A(X ): The set X is said to be a generating subset of the F -subset U 0 and of the subsystem A0 . A subsystem is said to be nitely generated if it admits a nite generating subset. We say that X U is a base subset (in other words, is a base or a base ) of an A-system A =(U f )if A is a generated set X and any proper subset X 0 X is not a generating set for A. If X is a base subset of an A-system A =(U F ), then we write U X ] and A X ] instead of U (X ) and A(X ), respectively. Note that every nitely generated A-system has a nite base. 1:4: Height and Length of Elements Let A(X ) = (U F ) be an A-system generated by a set X U . By the height of an element u 2 U with respect to X we mean the number hX (u) that is the least of the positive integers n for which u 2 Xn , where fXn g is the sequence (3) de ned in Subsection 1.3. It follows from the de nition that 1) hX (u) = 1 if and only if u 2 X 2) if hX (u)= n> 1, then the element u can be represented in the form u = f (u1 ::: uk ), where max(hX (u1 ) ::: hX (uk )) = n ; 1. Let us de ne the length lX (u)ofanelement u 2 U by induction on hX (u). If hX (u) = 1, then we assume that lX (u) = 1. If hX (u) > 1, then we consider all representations of u in the form u = f (u1 ::: un(f )) f 2F where hX (ui ) 1 1

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1:5: Indecomposability Condition for Elements An element u 2 U of an A-system A = (U F ) is said to be indecomposable if it cannot be represented in the form u = f (u1 ::: un(f ) ), where ui 6= u for at least one index i. By de nition, if an element u admits a representation of the form u = f (u : : : u), this is not a decomposition. Note that there are A-systems whichhave no indecomposable elements. For example, consider a set U of three elements x1 x2 x3 with a single binary operation (multiplication):

xi xi = x

i

i =1 2 3

xi xj = xk

for any pairwise distinct indices i j k. The following assertion results from the de nition of indecomposability.
able elements of this A-system.

Proposition 1.2. Each subset U

0

U generating an A-system (U F ) contains al l indecompos-

Corollary. If a subset X U of indecomposable elements generates the A-system (U F ), then X is a base, and this base of the A-system is unique.
1:6:N -Systems An A-system A =(U F ) is referred to as an N -system if it is generated by a subset X U of indecomposable elements. In this case, X is a base, and this base of the A-system A is unique in particular, A = A X ]. The cardinality of the set X is called the rank of the N -system A X ] and is denoted by

r(A X ]) = r(A)

r(A X ]) = #X:

Denote by h(u)and l(u) the height and thelengthofanelement u of an N -system with respect to its base.

a sequence of subsets Yn U by induction on n. Set Y1 = X10 \ U .Let Y1 ::: Yn;1 be already de ned. Let us de ne Y0n as the subset of elements in Xn \ U that do not belong to the subset U (Y1 Yn;1 )S U generated by Y1 Yn;1 . It follows from the de nition that the elements of the subset Y = 1 Yn are indecomposable in A0 =(U 0 F ) and generate A0 . n=1 In what follows, we consider only N -systems unless otherwise stated explicitly. 1:7: Commutative and Idempotent A-Systems An A-system A = (U F ) is said to be commutative if, for any operation f 2 F and any u1 ::: un(f ), the element u = f (u1 ::: un(f ) ) is preserved under all permutations of the elements u1 ::: un(f ) . An A-system A = (U F ) is said to be idempotent if f (u : : : u) = u for any u 2 U and any operation f 2 F . Note that if A =(U F ) is an idempotent A-system, then (fug F ) is a subsystem of A for any u 2 U.
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Proposition 1.3. Each subsystem A0 =(U 0 F ) of an N -system A X ]= (U F ) is an N -system. Proof. Let fXng be the sequence of subsets Xn U introduced0 in Subsection 1.3. We de ne 0


2. TAME A-SYSTEMS 2:1: Free A-Systems An A-system A(X )= (U F ) generated by a subset X U is said to be free if 1) the relation f (u1 ::: un(f ) ) = f 0 (u01 ::: u0n(f 0) ) holds if and only if f = f 0 (and hence n(f )= n(f 0 )= n) and ui = u0i, i =1 ::: n 2) no element u 2 X can be represented in the form u = f (u1 ::: un(f )). In particular, A is an N -system and X is its unique base consisting of indecomposable elements. The simplest example of a free A-system is given by a sequence fx1 ::: xn ::: g with the single unary operation f (xn )= xn+1 . The base of this A-system is X = fx1 g. The following proposition results from the de nition and Proposition 1.3. Proposition 2.1. Each subsystem of a free A-system is a free A-system. Let us note the following obvious fact. Proposition 2.2. In any free A-system A =(U F ), to each element x 2 U of height n, there corresponds an injective mapping x : F ! X (n+1) , where X (n+1) U is a subset of elements of height n +1 de ned by the formula x f = f (x : : : x):
and only if r(A) = r(A ) (i.e., #X = #X ) and #Fn = #Fn , n = 1 2 ::: , where Fn 0 Fn F 0 are the subsets of n-ary operations.

Proposition 2.3. Two free A-systems A X ]= (U F ) and 0A0 X 0]= (U 0 F 0 ) are isomorphic if 0 0

F and

Proof. In one direction, the assertion is obvious: if A-systems A and A0 are isomorphic, then the 0 condition of the proposition is satis ed. Conversely,let #X =#X 0 and #Fn =#Fn , n =1 2 :: : 0 and : Fn ! Fn , n =1 2 :: : Let us extend the bijection 0 then there exist bijections : X ! X to a bijection : U ! U 0 by induction on h(u). Let be already de ned for elements of heightless than n, and let h(u)= n, where n> 1. By de nition, the element u can be represented in the form u = f (u1 ::: uk(f )) f 2 F where h(ui ) 2:2: Embedding in a FreeGroupoid Consider a special case of a free A-system given by a free groupoid, i.e., a set with one binary operation whichwe call multiplication. According to the general de nition, a groupoid G generated by a subset X G is said to be free if 1) the relation x1 y1 = x2 y2 holds if and only if x1 = x2 and y1 = y2 2) no element x 2 X is representable in the form of a product x = x1 y1 .
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Let A X ]= (U F ) be a free A-system with base X U . Let us construct an embedding by induction on h(u), where G X then weset (u)= u.Let (u) be Let us rst de ne a mapping k of height less than n,by induction
1

(u)= (u)

U ! G X F] F ] is a free groupoid with base X F .If h(u)=1,i.e., u 2 X , already de ned for all elements of height less than n. :(Un ) k ! G X F ], where Un U is the subset of elements on k =1 2 ::: Namely,weset k (u1 ::: uk )= k;1 (u1 ::: uk;1 ) (uk ):

Note that the mappings k :(Un ) k ! G X F ] agree with the mappings k :(Um ) k ! G X F ] for m< n. Let h(u)= n. The element u has a representation in the form u = f (u1 ::: uk(f )), where f 2 F , h(ui ) RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 10 No. 1 2003


2) the following uniqueness condition holds for the decompositions: for any decomposable element u 2 U , there is a unique representation (up to the order elements ui ) of the form u = f (u1 ::: un(f )) where ui 6= u for at least one index i. In a tame A-system anytwo representations u = f (u1 ::: un(f )) and u = f (u01 ::: u0n(f )), (u1 ::: un(f ) ) and (u01 ::: u0n(f ) ) di er on the order only, are regarded as the same decompo It follows from the uniqueness condition for the decompositions that the following property Proposition 2.4. In a tame A-system, it follows from a decomposition

of the where sition. holds.

u = f (u1 ::: un)
that

where h(ui )
i =1 ::: n

h(u) = max(h(u1) ::: h(un)) + 1 l(u)= l(u1 )+ + l(un ): If an idempotent A-system is free, free commutative, or free idempotent, then it is a tame A-system. In general, in an arbitrary tame A-system, the permutation condition and the idempotent condition hold only for some subsets of operations and elements. Namely, let A X ] = (U F ) be an arbitrary tame A-system. For any f 2 F and u1 ::: un(f ) 2 U , write U (f )= fu 2 U j u = f (u : : : u)g
and denote by M (f u1 ::: un(f ) ) the subset of permutations (arrangements) u01 ::: u0n(f ) of the elements of u1 ::: un(f ) for which

f (u1 ::: u

n(f

)

)= f (u01 ::: u0n(f )):

The following assertion immediately results from the de nition of tame A-system. Proposition 2.5. Each tame A-system A X ]= (U F ) is uniquely de ned up to isomorphism by the sets U (f ) and M (f u1 ::: un(f ) ). Note that, for any tame A-system A = (U F ), one can de ne a natural surjection U 0 ! F , where U 0 U is the subset of all decomposable elements. Namely,to any element u 2 U 0 , there corresponds an element f 2 F de ning the decomposition of u, u = f (u1 ::: un ). Proposition 2.6. Every subsystem of a tame A-system is a tame A-system. As in the case of a free A-system, the assertion immediately follows from Proposition 1.3 and from the de nition of tame A-system. 2:5: Subordination Relation Let us introduce a subordination relation on elements of the support U of an arbitrary tame A-system A =(U F ). De nition. We say that an element u0 2 U is immediately subordinated to a decomposable element u, u 6= u0 , if one can nd a sequence fu1 ::: un g U containing u0 and an n-ary operation f 2 F such that u = f (u1 ::: un): It follows from the uniqueness of the decomposition that, for any decomposable element u 2 U , the set of elements immediately subordinated to u is nite and nonempty.
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or u is decomposable and there exists a nite sequence u = u1 u2 ::: un = u of decomposable elements in which each element except for the rst one is immediately subordinated to the preceding element. According to this de nition, if (U 0 F ) is a subsystem of (U F ), u0 u00 2 U 0 , and u00 is subordinated to u0 in the subsystem (U 0 F ), then u00 is subordinated to u0 in the A-system (U F ) as well. The following assertion also results from the de nitions. Proposition 2.7. If an element u0 is subordinated to an element u, u 6= u0 , then h(u0 ) 1, then l(u0) RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 10 No. 1 2003

De nition. Wesay that an element u0 2 U is subordinated to an element u 2 U if either u0 = u 0


Let us nowchoose an arbitrary element x 2 X and consider, for any n, the mapping Fn ! U taking every operation f 2 Fn to the element f (x : : : x) 2 U . This mapping is bijective. On the other hand, the elements f (x ::: x) are those and only those graph vertices from which exactly n edges issue, and all these edges end at the point x.Thus, for any n, the cardinality of the set Fn is also uniquely de ned by the graph G. 2:8: Subsets Xu Let X be the base of a tame A-system A =(U F ). Toanyelement u 2 U we assign the subset

Xu = fx 2 X j x 6 ug:
In particular, if u 2 X , then Xu = fug. The de nition implies the following assertion. Proposition 2.9. An element u belongs to the support U 0 U of a subsystem A X 0] with base 0 X if and only if Xu X 0 . X Proposition 2.7 implies the following assertion. Proposition 2.10. If u = f (u1 ::: un(f )), then

Xu = X X U be the base of U . Then
for any u 2 U 0 and Corollary. Let and x00 2 X 00 \ U 0 ,

u1

X

un(f

)

:
0

(4)

Proposition 2.11. Let A = (U F ) be a tame A-system, let U 0 0
h(u) >h(x)

U be an F -subset, and let

x 2 Xu , x 6= u, where l stands for the length of elements in U . U00 and00U 00 be F -subsets with 00ases X 0 U 0 0and X 00 U 00 . Then, if x0 2 X 0 \ U 00 b 0 0 x 6= x , then the relations x 2 Xx00 and x 2 Xx00 cannot hold simultaneously. Indeed, otherwise the relations h(x0 ) >h(x00 )and h(x00 ) >h(x0 )would hold simultaneously. Proposition 2.12. In the notation of the previous proposition, for any u 2 U 0 , we have l(u) >
X
x2Xu

l(x):

(5)

Moreover, if u = f (u1 ::: un(f ) ), where ui 2 U 0 , and if at least two sets Xui have nonempty intersection, then inequality (5) is strict.

Proof. (We proceed by induction on hX (u).) The assertion is obvious if hX (u)) = 1. Assume that hX (u)) = n > 1 then u = f (u1 ::: un(f ) ), where ui 2 U 0 , and hX (ui ) < hX (u) for any i. By the induction assumption, X l(ui) > l(x):
Therefore, the assertion for u immediately follows from (4) and from the relation
x2Xui

l(u)= l(u1 )+

+ l(un(f ) ):

Corollary 2.1. l(u) > #Xu. Corollary 2.2. If #Xu > 1, then l(u) >l(x) for any x 2 Xu.
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2:9: Completeness Condition Let U Y ] and U Z ]be F -subsets of a tame A-system, and let U Y ] U Z ]. Wesay that U Y ]is complete in U Z ]if U Y ] 6 U Z 0 ]for any proper subset Z 0 Z . The following assertion results from the de nition of the sets Xu . Proposition 2.13. U Y ] is complete in U Z ] if and only if Z = Zy :
y2Y

Proposition 2.14. If U Y ] U Z ], then U Y ] is a complete subset of U Z 0 ] U Z ], where
Z0 =
Moreover, if these subsets U Y ] and U Z ] are of nite rank, if r(U Y ]) > r(U Z ]), and if U Y ] is strictly containedin U Z ], then the set U Y ] is also strictly contained in U Z 0 ].
y2Y

Zy :

Proof. The rst assertion immediatelyfollows from the de nition of completeness. The other assertion is obvious if Z 0 = Z . If Z 0 is strictly contained in Z , then r(U Z 0 ]) < r(U Z ]). Thus, r(U Z 0]) l(z ):
Moreover, if at least two subsets Zy , y 2 Y , have nonempty intersection, then the inequality is strict.
y2Y z2Z

Proof. By Proposition 2.10,

l(y) >
X
y2Y

X
z2Zy

l(z ) l(z):

for any y 2 Y .Hence, Since

l(y) > Z=

XX
y2Y z2Zy y2Y

Z

y

by the completeness condition, this immediately proves the proposition. 2:10: Ascending Chain Condition Consider an arbitrary ascending sequence (not necessarily strictly ascending) of nite-rank F -subsets of a tame A-system, U Y1 ] U Yn ] (6) Proposition 2.16. If U Yn] is complete in U Yn+1 ] and r(U Yn]) > r(U Yn+1]) for any n, then the sequence (6) stabilizes at a nite step. Proof. Assume the contrary. Let there exist a strictly ascending sequence (6) satisfying the conditions of the proposition. By Proposition 2.15, X X l (y n ) > l(yn+1) for any n. Since, for this sequence, the ranks r( the sums yn 2Yn l(yn ) are stabilized at a nite step, we can assume that these ranks and sums are the same for all terms of the sequence.
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yn 2Yn yn+1 2Yn+1 U Yn ]) and

P


Consider an arbitrary pair Y 0 = Yn andP 00 = Yn+1 .Ifatleast two subsets Yy00 , y 2 Y 0 ,havea Y P nonemptyintersection, then y2Y 0 l(y ) > y2Y 00 l(y )(by Proposition 2.15), whichcontradicts the P assumption. Thus, the subsets Yy00 , y 2 Y 0 , are pairwise disjoint. In this case, since y2Y 0 l(y )= P 00 P 00 00 00 y2Y 00 l(y ) and Y = y2Y 0 ] Yy , we see that all sets Yy are singletons, Yy = fzy g moreover, 0 ! Y 00 .If l(y )= l(zy )for any y 2 Y 0 , then l(y ) > l(zy ), and the mapping y ! zy is a bijection Y y = zy , and therefore U Y 0 ]= U Y 00 ], whichcontradicts the assumption. If l(y) >l(zy ) for at least P P one y 2 Y 0 , then y2Y 0 l(y ) > y2Y 00 l(y ), which also contradicts the assumption. Theorem 2.1. For a tame A-system, every sequence (6) of F -subsets satisfying the inequality r(U Yn]) > r(U Yn+1]) for any n stabilizes at a nite step. Proof. Suppose the contrary. Let there exist a strictly ascending sequence (6) satisfying the conditions of the theorem. It follows from Proposition 2.13 that, in this case, there exists a strictly ascending sequence (6) such that U Yn ] is complete in U Yn+1 and r(U Yn ]) > r(U Yn+1 ]) for any n. This contradicts Proposition 2.16. 2:11: Decomposition Schemes Let A =(U F ) be a free A-system. To each element u 2 U we assign a directed graph S (u)of tree type we call this graph the decomposition scheme of the element u (or, brie y, the scheme of u). The vertex of this graph without incoming edges is called a root, and the vertices from which no edges issue are referred to as leaves. Let us de ne the graph S (u)by induction on the height h(u). If h(u) = 1, i.e., if u 2 X , where X is the base, then, by de nition, S (u) consists of one point which is the root and a leaf simultaneously.If h(u)= n> 1, then let us represent u in the form u = f (u1 ::: un ), n = n(f ), where h(ui ) < h(u), i = 1 ::: n. In this case, by de nition, the scheme S (u) is obtained from the schemes S (u1 ) ::: S (un)by adding one vertex (the root of the scheme S (u)) and n edges coming from this root to the roots of the schemes S (u1 ) ::: S (un). In this case, the root and the edges are equipped with labels, namely, the root has the label f (the symbol of the corresponding operation) and the edges are equipped with the digits 1 ::: n (the indices of the elements ui in the sequence (u1 ::: un )). This de nition of decomposition scheme still makes sense for free idempotent A-systems A = (U F ) because, for these systems, the decomposable elements u 2 U can also be represented uniquely in the form

u = f (u1 ::: un) where h(ui ) The de nition of decomposition schemes can also be extended to arbitrary tame A-systems. Here it is assumed that the labels at the edges contain additional information concerning the list of admissible permutations of elements in the expression u = f (u1 ::: un ). If the A-system in question is commutative, then we put no labels at the edges. In terms of schemes, the height h(u) of an element u is equal to the maximal number of tiers of the scheme S (u) and the length to the number of leaves of the scheme. Thus, if S (u)= S (v ), then h(u)= h(v ) and l(u)= l(v ). Example. Let A =(U F ) be the free A-system with a single binary operation (a free groupoid). In Figures 1, 2, and 3, we show the schemes of the elements (x1 x2 )(x3x4 ), (x1 (x2 x3 ))x4 , and x1 ((x2x3 )x4), respectively, where xi are arbitrary (not necessarily pairwise disjoint) elements of the base X U .

De nition. By S -subsets of a tame A-system A =(U F )we mean subsets V U formed by elements with the same decomposition scheme. According to this de nition, the set X of all indecomposable elements (i.e., elements of unit height) is an S -subset, and all other S -subsets are de ned by induction on the height of their elements. Namely, let the S -subsets consisting of elements of height less than k be already de ned. Then any S -subset V formed by elements of height k is given by an operation f 2 F and a
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1 1 1
Fig. 1

2 2

1 1

2 2 2

21 2

2

1 1

2

1

Fig. 2

Fig. 3

sequence of S -subsets V1 ::: Vn , where n is the arityof f , such that the heights of the elements of these subsets is less than k, and at least one of the subsets consists of elements of height k ; 1. The S -subset V is formed by the elements of height k that are representable in the form v = f (v1 ::: vn ), where vi 2 VI , i =1 ::: n. In particular, every S -subset of elements of height two is de ned by an operation f 2 F and consists of all elements of heighttwo representable in the form u = f (x1 ::: xn ), where xi 2 X , i =1 ::: n,and n is the arity of the operation f . Example. In any free groupoid the subset of all elements of heighttwoisan S -subset, as well as the subset of elements of height one. The family of elements of height three is decomposed into three S -subsets. Their representatives are elements of the form x1 (x2 x3 ), (x1 x2 )x3 , and (x1 x2 )(x3 x4). Denote by Hn the number of S -subsets of a free groupoid formed by elements of height n. It follows from what was said abovethat H1 = H2 =1and H3 = 3. Let us show that the following recurrence formula holds for any n> 2.

H Hn = Hn;1 + Hn;1 + Hn;2 Hn;1 : n;2 Indeed, there are three types of S -subsets with elements of height n,namely, the S -subsets with elements of the form xy , where 1) h(x)= h(y )= n ; 1, 2) h(x)= n ; 1 and h(y ) < n ; 1, and 3) h(x) 2 by the fol lowing recurrence formula : H Hn = Hn;1 + Hn;1 + Hn; 2 n;2
In particular, H3 =2, H4 =7, H5 =112, etc.
2

Assertion.

Hn;1 :

2:12:A-Systems of S -Subsets Toany tame A-system A =(U F ) we assign another A-system AS =( F ) whose support is the family of all S -subsets of U and the fundamental set coincides with the fundamental set of the original A-system.
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The action of the operations f 2 F on the set is introduced in the natural way. Namely, let f 2 F be an arbitrary n-ary operation, V1 ::: Vn are arbitrary S -subsets of U , and vi 2 Vi, i =1 ::: n, are some representatives of the sets Vi. It is assumed that, if some sets Vi coincide, then their representatives coincide as well. By de nition, V = f (V1 ::: Vn )isthe S -subset containing the element v = f (v1 ::: vn ). The set V is well de ned because it does not depend on the choice of the representatives vi 2 Vi . It follows from the de nition that (1) the A-system AS of the S -subsets of a tame A-system A is also tame (2) if A is a free, free commutative, or a free idempotent A-system, then the A-system AS of the S -subsets of A is also free, free commutative, or free idempotent A-system, respectively (3) if A is a free or a free commutative A-system, then the base of the A-system AS consists of a single element, namely,of the S -subset X of elements of unit height (4) if A is a free idempotent A-system, then the S -subset V belongs to the base of the A-system AS (i.e., is an indecomposable elementofthis A-system) if and only if either V = X or the elements v 2 V are of the form v = f (v1 ::: vn ), where h(vi )
ui = f (ui1 ::: uin ) where uij 2 Vj : By the induction assumption, the corresponding element vj = '(u1j ::: umj ) 2 Vj is already de ned for any j , j =1 ::: n. Set '(u1 ::: um)= f (v1 ::: vn ): Obviously, '(u1 ::: um ) 2 V . Example. If X is endowed with the structure of a group G, then this structure induces a group structure on any S -subset V U . The group thus obtained is isomorphic to the direct product of l copies of the group G, where l is the length of the elements in V .
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3. LATTICES RELATED TO TAME A-SYSTEMS 3:1. Intersections and Unions of F -Subsets Let A X ]= (U F ) be an arbitrary tame A-system with base X U (the subset of indecomposable elements). Consider the family of F -subsets U 0 U , i.e., the subsets closed with respect to the operations f 2 F . Recall that we denote by U Y ] the F -subset of U with base Y . By Proposition 2.6, Y coincides with the set of indecomposable elements in U Y ]. Denote by r(U Y ]) the rank of this subset, i.e., r(U Y ]) = #Y . Proposition 3.1. For any Y X and Z X , U Y ] \ U Z ]= U Y \ Z ]:

Proof. Obviously, U Y \ Z ] U Y ] \ U Z ]. Conversely,if u 2 U Y ] \ U Z ], then Xu 2 Y \ Z , and therefore u 2 U Y \ Z ]. Theorem 3.1. The intersection of arbitrary F -subsets U Y ] and U Z ] is generated by the subset Y 0 Z 0 , where Y 0 = Y \ U Z ] and Z 0 = Z \ U Y ]. Thus, U Y ] \ U Z ]= U W ] where W is a subset of the set Y 0 Z 0 . In particular, if Y 0 = Z 0 = ?, then U Y ] \ U Z ]= ?. Proof. Write B = U (Y 0 Z 0 ). It is clear that B U Y ] \ U Z ]. Assume that U Y ] \ U Z ] 6 B. Choose an element u 2 (U Y ] \ U Z ]) n B of the minimal possible height hY (u). Since hY (u) > 1 and hZ (u) > 1, the element u can be represented in the form u = f1 (a1 ::: am )= f2 (b1 ::: bn ) where ai 2 U Y ], hY (ai ) (Y n Y 0 ) (Z n Z 0 ) (Y 0 \ Z 0 ) for any subsets Y 0 Y \ U Z ] and Z 0 Z \ U Y ].Hence,

W

U1 _ U2 = U ((Y n Y 0 ) i.e., U1 _ U2 is generated by the subset (Y n Y 0 ) Proof. Write Y n Y 0 = Y and Z 0 n Y 0 = Q. z 2 U (Y (Z nfz

Z 0 ) (Y 0 \ Z 0 )) Z 0 ) (Y 0 \ Z 0 ). rst prove that g)) for any z 2 Q:
(Z n (Z n Let us
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(7)

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Let z 2 Q and y 2 Yz \ Y 0 . Since y 6= z , it follows from Proposition 2.11 z 2 Zy , i.e., y 2 U Z nfz g]. Hence, Yz \ Y 0 U Z nfz g]. Therefore, since = follows that Yz U (Y (Z nfz g)) for any z 2 Q: This proves relation (7). It follows from (7) that U Z ] U (Y (Z nfz g Y 0 U Z ], wealso have U Y ] U (Y (Z nfzg)). Thus, U (Y Z )= U (Y (Z nfz g)) for any z 2 Q and hence W Y (Z nfz g) for any z 2 Q, which implies W Y note that Y (Z n Q)= (Y n Y 0 ) (Z n Z 0 ) (Y 0 \ Z 0 ): This completes the proof of the theorem.

(see the corollary) that Yz (Yz \ Y 0 ) Y ,it )) for any z 2 Q. Since (Z n Q). It remains to

U (Y Z )= U ((Y n Y 0 ) Z )= U (Y (Z n Z 0 )) = U ((Y n Y 0 ) (Z n Z 0 ) (Y \ Z )) where Y 0 = Y \ U Z ] and Z 0 = Z \ U Y ]. 3:2: Latticeof F -Subsets of Finite Rank Denote by L = L(U F ) the family of F -subsets of nite rank in a tame A-system A =(U F ). The set L forms a lattice with respect to the above operations of taking the sum (union) _ and the product (intersection) ^. According to 3.1, if U1 = U Y ]and U2 = U Z ], then U1 _ U2 = U (Y Z ), U1 ^ U2 = U W ], where W (Y \ U Z ]) (Z \ U Y ]). Thus, the operations of union and intersection over F -subsets in L are reduced to related operations over their bases. Theorem 3.3. The ranks of the F -subsets U1, U2, U1 ^ U2 ,and U1 _ U2 are related as follows : r(U1 ^ U2 )+ r(U1 _ U2 ) 6 r(U1 )+ r(U2): (8) Proof. Let U1 = U Y ], U2 = U Z ], and U1 ^ U2 = U W ]. Since W (Y \ U Z ]) (Z \ U Y ]), let us represent W in the form of a disjoint union W = Y 0 Z 0 , where Y 0 Y \ U Z ] and Z 0 Z \ U Y ]. By Theorem 3.2, then wehave U1 _ U2 = U (Y Z )= U ((Y n Y 0 ) (Z n Z 0 )): Hence, r(U1 _ U2 ) 6 #(Y n Y 0 ) (Z n Z 0 )]: Since #(Y n Y 0 ) (Z n Z 0 )] 6 (#Y )+ (#Z ) ; (#(Y 0 Z 0 )) it follows that r(U1 _ U2 ) 6 r(U1 )+ r(U2) ; r(U1 ^ U2 ). Note. There are examples of nite-rank F -subsets U1 and U2 satisfying the inequalities r(U1 ^ U2 ) > max(r(U1 ) r(U2)) and r(U1 _ U2 ) < min(r(U1) r(U2)): For instance, let U be the free groupoid generated by some elements x and y . Let us de ne the elements xn and yn by induction on n, x1 = x y1 = y xn+1 = xn x yn+1 = yn y n =1 2 :: : Let U1 and U2 be the subgroupoids generated by the elements y x1 ::: xm and x y1 ::: yn , respectively. Then U1 _ U2 = U , i.e., the groupoid U1 _ U2 is generated by the elements x and y , and the groupoid U1 ^ U2 is generated by the elements x1 ::: xm y1 ::: yn .Hence, r(U1 )= m +1, r(U2)= n +1, r(U1 _ U2 ) = 2, and r(U1 ^ U2)= m + n. Note that, in this example, wehave r(U1 ^ U2)+ r(U1 _ U2 )= r(U1 )+ r(U2): The lattice L is \not geometric" because, in this lattice, the relation U1 U does not imply the inequality r(U1 ) 6 r(U2 ). Moreover, in general, F -subsets of nite rank can contain F -subsets of any nite rank and even F -subsets of in nite rank. This lattice is not semimodular. For instance, in the free idempotent groupoid G x y ], the one-element subgroupoids G1 = G x]and G2 = G y ] cover (in the sense of lattice theory) the subgroupoid G3 = ?.However, their union G x y ] does not cover G1 and G2 .For instance, the subgroupoid G x xy ]contains G1 and is contained in G x y ].
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Corollary.


3:3:F -Subsets of Finite Corank and Their Cobases Let A =(U F ) be an arbitrary tame A-system. De nition. We say that an F -subset V U has nite corank if one can nd nitely many elements u1 ::: un 2 U such that the F -subset of U generated by the subset V and the elements u1 ::: un coincides with U . The minimal number r of these additional elements is called the corank of the F -subset V andisdenoted by cor(V ). Note that, if U is of nite rank n, then all nonempty F -subsets V U have nite corank not exceeding the number n ; 1. Proposition 3.2. If X is the base of the support U of an A-system, then an F -subset V with base Y has nite corank r if and only if Y contains al l elements of the base X of U possibly except for nitely many elements, i.e., the set X n (X \ Y ) is nite. Proof. Obviously, if Y contains all elements of X except for nitely many, then the F -set V has nite corank. Conversely, let V be an F -set of nite corank r. Then there exist r elements u1 ::: ur 2 U such that the F -subset generated by V and these elements coincides with U . Hence, the base X of the set U is contained in Y fu1 ::: ur g,i.e., X Y fu1 ::: ur g.Thus,

X =(X \ Y ) (Y \fu1 ::: ur g): (9) In other words, Y contains all elements of the base X except for nitely many elements of X . Proposition 3.3. For any F -subset V of corank r,there exists an r-tuple of elements u1 ::: ur in X such that the F -subset generated by the set V and the elements u1 ::: ur coincides with U . This r-tuple is uniquely de ned and is given by fu1 ::: urg = X n (X \ Y ) (10) where Y is the base in V . The set of elements u1 ::: ur de ned by relation (10) is called the cobase of the F -subset V . Proof. Let us apply formula (9). Note that all elements ui belong to X . Indeed, if, for instance, ur 2 X , then X = (X \ Y ) (Y \fu1 ::: ur;1g) according to (9). Therefore, U is generated = by the set Y and the elements u1 ::: ur;1 , which contradicts the condition that the number r is minimal possible. Thus, it follows from (9) that X = (X \ Y ) fu1 ::: ur g, and therefore fu1 ::: urg = X n (X \ Y ), which proves the proposition. Note that distinct F -subsets can have equal cobases. For instance, in the free groupoid with base fx y g, all F -subsets of rank one such that the base element of this subset di ers from x and y have the set fx yg as the cobase.
3:4: Planar Subsets Let A =(U F ) be a tame A-system. Introduce the family L of all nite-rank F -subsets. Later on, we shall see that this family forms a semimodular lattice. De nition. An F -subset U0 0 U of nite rank r is said to be a planar subset if there exists no F -subset strictly containing U that is of rank r1 6 r. The following assertion results from the de nition. Proposition 3.4. If U 0 = U Y ] is a planar subset of rank r, then (1) r(U 00 ) >r(U 0 ) for any F -subset U 00 strictly containing U 0 (2) r(U 00 ) RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 10 No. 1 2003


3:5: Planar Envelopes of an F -Subset Proposition 3.5. For any F -subset U 0 of nite rank r, there exists a planar subset of rank r1 6 r containing U 0. The assertion immediately follows from Theorem 2.1 concerning the ascending chain condition for F -subsets. By the reduced rank of a nite-rank F -subset U 00 we mean the least positiveinteger r for0 which there exists a planar subset of rank r containing U . Denote the reduced rank of U 0 by p(U ). The following assertions result from the de nition: (1) p(U 0 ) 6 r(U 0 ) (2) if U 0 is a planar subset, then p(U 0 )= r(U 0 ) (3) r(U 00 ) > p(U 0 ) for any F -subset U 00 U 0 . Lemma 3.1. Let U1 = U Y ] and U2 = U Z ] be planar subsets of ranks k1 and k2, respectively, where k1 6 k2 , and let n be the rank of their intersection V = U1 \ U2 . Then (1) n 6 k1 (2) if n = k1 , then V = U1 , i.e., U1 U2 (3) if n = k1 = k2 , then V = U1 = U2 . Proof. Consider the F -subset B = U (Y Z ). Wehave B U2 , and hence r(B) 6 k1 + k2 ; n by Theorem 3.3. If n> k1 , then r(B ) unique.

Then V U , and therefore r(V ) > p(U ). However, r(P1 ) = r(P2 ) = p(U ), and therefore the relation P1 = P2 follows from Lemma 3.1. De nition. The planar subset P de ned by Theorem 3.4 is called the planar envelope of the F -subset U 0 , or, in other words, the planar subset generated by U 0 . 3:6: Properties of Planar Subsets Theorem0 3.5. If a planar subset P contains an F -subset U 0 , then P contains the planar envelope P 0 of U . Proof. We have r(P 0 ) = p(U 0) and r(0 P ) > p(U 0). Let V = P00 \ P . Then 0 V U 0 . Hence, r(V ) > p(U 0). By Lemma 3.1, r(V ) 6 r(U ). Therefore, r(V )= p(U ). Since r(P \ P )= r(P 0 ), it follows from Lemma 3.1 that P 0 P ,aswas to be proved. Theorem 3.6. Each intersection V = \ P of planar subsets P of ranks r is a planar subset of rank r 6 min r . In particular, if r = r for some , then V = P . Proof. By Theorem 3.4, there exists a unique planar subset P V generated by V . By Theorem 3.5, P P for any , and so P V . Hence, P = V and r(V )= r(P )= p(V ) 6 min r . If r = r for some , then V = P by Lemma 3.1. Theorem 3.7. If P1 , P2, and P3 are F -subsets such that P1 is a planar subset of P2 and P2 is a planar subset of P3 , then P1 is a planar subset of P3 . Proof. It follows from the condition of the theorem that r(P1) 6 r(P2) 6 r(P3). Let P be the planar subset of P3 generated by P1 . Then r(P ) 6 r(P1 ). By the previous theorem, P2 \ P is a planar subset of P3 . Since r(P ) 6 r(P2 ), it follows that r(P2 \ P ) 6 r(P ) 6 r(P1 ). Thus, together
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Proof. Let P and P be planar subsets of rank p(U 0) that contain U 0, a0nd let V = P \ P . 0 0
1 2 1 2


with the de nition of planar subset, implies that P2 \ P = P1 , and therefore Then r(P2 \ P )= r(P ) and P P2 by Theorem 3.6. Thus, P = P1 . Let us give a su cient condition for an F -subset U Y ] U of nite rank n A =(U F ) with base X U to be planar. Proposition 3.6. If the subsets Xy , y 2 Y , are pairwise disjoint and if y 2 Y , then the F -subset U Y ] is planar. Proof. Suppose the contrary. Let U Y ] be strictly contained in an F -subset Consider the subset Zy Z , y 2 Y .Wehave

r(P2 \ P )= r(P1 ).
in a tame A-system

l(y ) = #Xy for all U Z ]ofrank r 6 n.

Xy =

z2Zy

X

z

hence, if at least two subsets Zy havenonemptyintersection, then the corresponding subsets Xy also have nonemptyintersection, whichcontradicts the assumption. Thus, the subsets Zy are pairwise disjoint. Since r 6 n, this can occur only if r = n. In this case, all subsets Zy are singletons, i.e., Zy = fzy g, where zy 2 Z , and the mapping y 7! zy is a bijection of Y onto Z . Note that zy 6= y for at least one y 2 Y because otherwise the F -subsets U Y ]and U Z ] coincide. If zy 6= y , then l(y ) > l(zy ). On the other hand, Xy = Xzy . Therefore, since l(zy ) > #Xzy , it follows that l(y ) > #Xy , which contradicts the condition. 3:7: Lattices of Planar Subsets Denote by L = L(U F ) the family of planar subsets of a tame A-system A =(U F ). Since the intersection of planar subsets is a planar subset, it follows that L is equipped with the structure of a lattice by inclusion: the product U1 ^ U2 of planar subsets U1 = U Y1 ] and U2 = U Y2 ] is de ned as in the lattice L, and the sum U1 _ U2 is the planar subset generated by the F -subset U (Y1 Y2 ). Since the de nitions of the sum on L and L are distinct, L is not a sublattice of L. Theorem 3.8. The ranks of any two planar subsets U1 and U2 and those of the planar subsets U1 ^ U2 and U1 _ U2 are related by inequality (8). This inequalityfollows from a similar inequality in the lattice L if one takes into account that the rank of the sum of subgroupoids in the lattice L does not exceed the rank of their sum in the lattice L. De nition. Wesay that a planar subset V1 covers a planar subset V2 if U1 strictly contains U2 and there exists no planar subset V distinct from V1 and V2 and such that V1 V V2 . Theorem 3.9. If V U is an arbitrary planar subset and U 6= V , then any planar subset = V 0 covering V is generated by V and by some element u 2 V . Conversely, any planar subset V 0 generated by V and by an element u 2 V covers V . = Proof. Let V 0 V , let x 2 V 0 n V , and let V 00 be a planar subset generated by V and u. Then 0 V 00 V and V 00 6= V . Therefore, if V 0 covers V , then V 0 = V 00 . V Conversely,if V 0 is a planar subset generated by V and by some element u 2 V , then r(V ) < = r(V 0 ) 6 r(V ) + 1, and thus r(V 0 )= r(V ) + 1. Hence, V 0 covers V . Theorem 3.10. The lattice L of the planar subsets satis es the semimodularity condition 4]: if planar subsets U1 and U2 , U1 6= U2 ,cover a planar subset U0 ,then U1 _ U2 covers both U1 and U2 . Proof. Let r(U )= n. In this case, r(U1)= r(U2) = n + 1. It follows from Theorem 3.8 that r(U1 _ U2) 6 n + 2. On the other hand, since U1 6= U2,it follows that the subset U1 _ U2 strictly contains U1 and U2 , and therefore r(U1 _ U2 ) >n + 1. Hence, r(U1 _ U2 )= n +2.
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G x y z] G x yz] G x] G y xz ]

?
Fig. 4

Note that the lattice L can be not modular. For instance, the lattice L related to the free groupoid G x y z] contains the sublattice shown in Fig. 4. However, modular lattices have no sublattices of this form. 3:8: Geometric Structures on Tame A-Systems Since the lattice L of all planar subsets is semimodular, it follows that the support U of a tame A-system can naturally be treated as a pro jective space, and the planar subsets of rank r by themselves canbeviewed as (r ; 1)-dimensional planes in this space. In particular, the planar subsets of ranks one and two will be referred to as points and lines, respectively. This point of view is especially convenient in the case of a free idempotent A-system because, for this system, all planar subsets of rank one are singletons. However, as in the case of an arbitrary tame A-system, one can replace every planar set of rank one (i.e., a \pro jectivepoint") by an element u 2 U generating this set. In the geometry thus arising, the main axioms concerning the unions and intersections of planes in pro jective spaces are satis ed. For instance, one and only one line can be drawn to pass through two distinct points. Every line can be disjoint from a plane, can have exactly one pointofintersection with this plane, or can belong to the plane. Exactly one two-dimensional plane passes through two (noncoinciding) intersecting lines, etc. The speci c feature of this geometry is that every k-dimensional plane is uniquely equipped with the base formed by(k +1) points, i.e., the set of indecomposable elements. Moreover, the base of the intersection of two planes is contained in the union of the bases of these planes. For this reason, for instance, the family of lines passing through a chosen point u is partitioned into two subsets formed by the lines that contain or do not contain the point u in the set of their base points. If points a, b,and c form the base of a two-dimensional plane, then any line passing through two of these three points has these points as the base. If a line l on the plane intersects the lines a b and a c, then the points of intersection form the base of the line l. Hence, l cannot intersect the line b c. 3:9: Planes De nition. An F -subset V U of a tame A-system A =(U F ) is called a plane if any subset in V is a planar subset in U . In particular, the support U of the A-system and the emptyset ? are planes. The following assertions result from the de nition. (1) The notion of plane is transitive: if V1 is a plane in U and V2 is a plane in V1 , then plane in U . (2) Every planar subset in U is a plane and, conversely,every plane of nite rank is a subset. In particular, if the rank of the support U is nite, then every plane in U is a subset. (3) Every plane V U is the union of all planar subset in U belonging to V . Theorem 3.11. The intersection V = T V of any family of planes V is a plane.
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planar

V2 is a planar planar

2003


V be an arbitrary planar subset in V , i.e., a minimal planar subset of V containing planar subset of U , and hence so is the intersection \ V 00 = V

Proof. Let V

0

of V and let V 0 be a planar covering of V 0 V 0 . By the de nition of plane, every V 0 is a
0:

Let us prove that V 0 = V 00 . Indeed, since r(V 0 ) 6 r(V 0 )for any , it follows that r(V 00 ) 6 r(V 0 ). Since V 0 is a planar subset of V and V 0 V 00 V , this inequality is possible only if V 00 = V 0 . Corollary. For any subset A U , there exists the smallest plane containing this subset, namely, the intersection of al l planes in U containing A. De nition. The smallest plane in U containing a subset A U is called the planar envelope or the planar covering of A.Introduce an operation on the family of subsets of U by de nition, A B is the planar covering of the set A B . The operation is commutative, associative, and also has the following properties. (1) If (A B ) (A0 B 0 ), then A B A0 B 0 . In particular, if (A B )=(A0 B 0 ), then A B = A0 B0 if B A, then A B = A ?. (2) A A B = A B for any subsets A and B . (3) If V is a plane, then V V = V . (4) For any planes V and W in U ,wehave V _ W = V W = (V 0 W 0 ) where the union is taken over the family of all planar subsets V 0 and W 0 contained in V and W , respectively. Let us present a criterion for a subset V U to be a plane (in terms of the operation ). Proposition 3.7. A subset V U is a plane in U if and only if V is the union of some planar subsets in U and, for any planar subsets V 0 and V 00 contained in V , the planar subset V 0 V 00 is also containedin V . Proposition 3.8. For any plane V U and any element x 2 V , we have = 0 fxg) V fxg = (V where the union is taken over al l planar subsets V 0 in U contained in V . Proof. Write W = (V 0 fxg). For any planar subsets V 0 and V 00 in V ,wehave e e (V 0 fxg) (V 00 fxg)= (V fxg) where V = V 0 V 00 : e Since V is a planar subset contained in V , it follows that the set W satis es the conditions of Proposition 3.7, and therefore it is a plane in U . Since0 V W and x 2 W , it follows that V fxg W . On the other hand, W V fxg because V fxg V fxg for any planar subset V 0 V .Thus, V fxg = W . De nition. Wesay that a plane V1 covers a plane V2 if U1 strictly contains U2 and there exists no plane V distinct from V1 and V2 and such that V1 V V2 . Theorem 3.12. A plane V 0 covers a plane V U , V 6= U , if and only if V 0 = V fxg, where x 2 V . Thus, the set planes covering a plane V 6= U is not empty. = Proof. 0 In one direction the assertion is clear, namely, if a plane 0V 0 strictly contains a plane V and x 2 V n V , then V 0 V fxg V and V fxg6= V .Thus, if V covers V , then V 0 = V fxg. Conversely, let us prove that a plane V fxg, where x 2 V , covers the plane V . Suppose the = contrary. Let there exist a plane W distinct from V and V fxg and such that V fxg W V . Let u 2 W n V . Then V fug6= V , V fug6= V fxg,and V fxg V fug V . By Proposition 3.8, there exists a planar subset V 0 V such that u 2 V 0 fxg, and hence 0 fxg V 0 fug V 0 . By Theorem 3.10, the planar set V 0 fxg covers the planar subset V 0 : V Therefore, since V 0 fug6= V 0 fxg it follows that V 0 fug = V 0 i.e., u 2 V 0 which is not the case.
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3:10: Lattice of al l Planes e It follows from Theorem 3.11 that the set L of all planes in the support U of a tame A-system A =(U F ) is equipped with the structure of a lattice with respect to embedding. In this lattice, for any planes V1 and V2 , the element V1 ^ V2 is de ned as the intersection of these planes and V1 _ V2 as the smallest plane containing V1 and V2. The family L of planar subsets is a sublattice of this lattice. Here are other examples of sublattices e in L : 1) the sublattice of planes of countable rank, 2) the sublattice of planes of nite corank. Remark. The family of planes V U such that V fu1 ::: ung = U for an appropriate nite subset fu1 ::: un g U is closed with respect to union but not closed with respect to e intersection therefore, it does not form a sublattice of L. For instance, in the free groupoid with the base fx y z1 ::: zn ::: g, the F -subsets V1 and V2 with the bases fx y z1 ::: y zn ::: g and fy xz1 ::: xzn ::: g are planes. The relations V1 fyg = V2 fxg = U hold however, V1 \ V2 = ?. We stress that V1 and V2 have in nite corank and that their intersection is the emptyset. covers both U1 and U2 . It follows from Theorem 3.12 that there exist elements x1 x2 2 V such that V1 = V fx1 g and = V2 = V fx2g,where x1 6= V fx2 g and x2 6= V fx1 g. Then the plane V1 _ V2 = V fx1g fx2 g covers V1 and V2 because V fx1 g fx2 g = V1 fx2 g = V2 fx1 g. 3:11: Inductive Limits of Tame A-Systems and RelatedLattices of Planes Let us extend the class of A-systems passing from the tame A-systems to their inductive limits. Assume that a family of tame A-systems A =(U F )is given, where the index ranges over a partially ordered set in which, for any 2 , there exists a 2 for which < and < . Further, we assume that the following ob jects are de ned for any ordered pair of indices where < : 1) an injection (embedding) : U ,! U 2) bijections : Fn ! Fn n =1 2 :: : where Fn is the subset of the n-ary operations. It is assumed that = = for any ordered triple of indices < < . By 2), one can assume that any set F is identi ed with a chosen set F . Sets f g and f g are said to be compatible if the following properties hold for any ordered pair , where < : a) U U is an F -subset b) (f (u1 ::: un )) = ( f )( u1 ::: un )for any n =1 2 :: : ,any operation f 2 Fn , and any elements u1 ::: un 2 U . Toany compatible system f g and f g one can assign the inductive limit A = lim ind A , whichisan A-system A =(U F )with U = lim ind U . De nition. An A-system A = lim ind A is said to be weakly tame if, for any ordered pair , where 6 , the subset U U is a plane in U .
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e Theorem 3.13. The lattice L is semimodular. Proof. It su ces to prove that if planes U and U , U 6= U ,cover the plane U , then U _ U
1 2 1 2 0 1 2


Obviously,every tame A-system is weakly tame. For example, the inductive limit of a sequence groupoids G Xn ], n = 1 2 ::: , with the bases Xn = fxn 1 ::: xn 2n; g is a weakly tame A-system. The embedding G Xn] ,! G Xn+1 ]isgiven by the formula xni = xn+1 2i;1 xn+1 2i i =1 ::: 2n;1: De nition. Let A = (U F ), where U = lim ind U , be an arbitrary weakly tame A-system. A subset V U is called a plane if the intersection V = V \ U is a plane in U for any index . According to this de nition, every plane V U is the inductive limit of the planes V = V \ U U , V =lim ind V . Obviously,any plane is an F -subset of U . It follows from Theorem 3.11 that the intersection of each family of planes of a weakly tame A-system is also a plane. Thus, the set of planes of a weakly tame A-system is equipped with the structure of a lattice with respect to the operation of embedding. The notion of planar coverings of subsets and the operation can naturally be extended to the weakly tame A-systems. Note that, for every plane V = lim ind V and anypoint x 2 V ,wehave = the relation V fxg = lim ind V fxg:
1

De nition. As in the case of tame A-systems, we say that a plane V1 of a weakly tame A-system covers a plane V2 if U1 strictly contains U2 and there exists no plane V distinct from V1 and V2 and such that V1 V V2 . Theorem 3.14. A plane V 0 of a weakly tame A-system A = (U F ), where U = lim ind U , covers a plane V U , V 6= U , if and only if V 0 = V fxg, where x 2 V . Thus, the set of planes = covering a plane V 6= U is not empty. Proof. If the plane V 0 strictly contains the0 plane V and x 2 V 0 n V , then V 0 V fxg V and V fxg6= V .Thus, if V 0 covers V ,then V = V fxg. Conversely, let us prove that the plane V fxg,where x 2 V ,covers the plane V . Assume the = contrary. Let there exist a plane W distinct from V and V fxg and such that V fxg W V . Let u 2 W n U . Then V fxg V fug V , where the planes V fxg, V fug, and V are pairwise distinct. Since V fxg = lim ind V fxg, where V = V \ U , it follows that there is an index 0 for which x 2 U and u 2 V fxg for any > 0 . Thus, V fxg V fug V for any > 0 . By Theorem 3.12, the plane V fxg covers V . Since V fug6= V , this implies that V fxg = V fug for any > 0. Then V fxg = V fug, whichcontradicts the assumption. Theorem 3.15. The lattice of planes of a weakly tame A-system is semimodular. The proof is just the same as in Theorem 3.13.
4. TOPOLOGICAL STRUCTURES ON A-SYSTEMS AND ON THE SETS OF THEIR SUBSYSTEMS 4:1: Topological A-Systems Let us de ne twotypes of topological A-systems, namely, A-systems A =(U F ) for which the topology is de ned on the support U only and A-systems for which both the support U and the fundamental set F are equipped with some topologies. De nition. An A-system A =(U F )is calledan AT -system if the support U is equipped with the structure of a topological space with respect to which the mappings (u1 ::: un ) 2 U n ! f (u1 ::: un ) 2 U are continuous for any n and any f 2 Fn .
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De nition. An A-system A =(U F )iscalledan AT F -system if U and F are equipped with the structures of topological spaces such that the mappings
(f u1 ::: un) 2 Fn U n ! f (u1 ::: un ) 2 U are continuous with respect to the topologies for any n. If the topology on F is discrete, then these de nitions are equivalent. Every topology on the support U of the A-system A = (U F ) induces some topology on the family L = L(U F ) of all nitely generated subsystems of this A-system or, equivalently, on the family of nitely generated F -subsets U 0 U . This topology is de ned as follows. Denote by M (Vy ::: Vyn ), where Vyi U are some neighborhoods of the points y1 ::: yn , respectively, the family of all F -subsets U (z1 ::: zn ) generated by the elements zi 2 Vyi , i =1 ::: n. De nition. Let us introduce a topology on L = L(U F ) as follows. For a base of neighborhoods of any nitely generated F -subset U 0 U ,we take the family of sets M (Vy ::: Vyn ), where the collection Y = fy1 ::: yn g ranges over the bases in U 0 U and Vy ::: Vyn range over the bases of neighborhoods of the elements in Y . This de nition can be simpli ed for N -systems because, for an N -system, every nitely generated F -subset U 0 U has a unique base Y = fy1 ::: yn g. This implies the following assertion. Proposition 4.1. A Hausdor topology on the support U of an N -system induces a Hausdor topology on the families Ln U of al l F -subsets U 0 U of arbitrarily chosen rank n.
1 1 1

4:2: Free AT -Systems Let A X ]= (U F ) be a free A-system, and let the base X U of this system be equipped with the structure of a topological space, i.e., a base of neighborhoods of any point x 2 X is de ned. Let us construction the extension of the topology on X to a topology on U with respect to which all operations f 2 F are continuous. To this end, let us de ne the bases of neighborhoods of all points u 2 U by induction on the height h(u). If h(u) = 1, i.e., u 2 X , then the base of neighborhoods Vu X of the point u is (originally) de ned. Let the bases of neighborhoods be already de ned for all points of heightless than n, and let h(u)= n> 1. Then u can be represented (and this representation is unique) in the form u = f (u1 ::: um) where h(ui ) < n, i = 1 ::: m, and hence the bases of neighborhoods of the elements ui have already been de ned. Let us de ne the base of neighborhoods of the element u as the family of sets
0 f (Vu ::: Vum )= fu0 = f (u01 ::: u0m ) j u01 2 Vu ::: um 2 Vum g
1 1

where Vu ::: Vum range over the bases of neighborhoods of the elements u1 ::: um . It follows from the de nition that all operations f 2 F are continuous with respect to the topology thus introduced. According to Subsection 4.1, we refer to an A-system A X ] = (U F ) with the topology on U thus de ned as a free AT -system. In what follows, it is assumed that the topology on X is Hausdor . Then the topology on U is also Hausdor . It follows from the de nition of the topology on U that the neighborhoods f (Vu ::: Vum ) of the element u = f (u1 ::: um ) are homeomorphic to the Cartesian product Vu Vum . Hence, applying the usual induction on the heightofelements, we obtain the following assertion.
1 1 1

Proposition 4.2. The properties of local connectedness and local compactness are preserved under the extension of the topology from X to U .
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Proposition 4.3. Any S -subset US scheme S ) is open and closed.

U (i.e., a subset with an arbitrarily xed decomposition

Proof. It su ces to prove (by induction on the height) that, for any u 2 U , there exists a neighborhood all of whose elements have the same decomposition scheme as that of u. In the case of h(u) = 1, the assertion is obvious. If h(u) = n > 1, then let us represent u in the form u = f (u1 ::: um), where h(ui) < n. By the induction assumption, for any ui, there exists a neighborhood Vui all of whose elements have the same decomposition scheme as that of ui . Then all elements u0 in the neighborhood f (Vu ::: Vum )of u have the same decomposition scheme as that of u. Corollary 4.1. The subsets of elements u 2 U of any xed height and the subsets of elements
1

of any xed length open and closed. Corollary 4.2. All F -subsets of nite rank are discrete.

(This holds because these sets contain only nitely many elements of anychosen height.) Proposition 4.4. Let A X ]= (U F ) be a free AT -system with base X U , let y1 ::: yn be arbitrary elements in U (not necessarily pairwise distinct ), and let Vy ::: Vyn be any neighborhoods of these elements in U . Then, if X contains no isolated points, then there exist elements zi 2 Vyi i =1 ::: n such that the sets Xzi are pairwise disjoint and l(zi)=#Xzi , i =1 ::: n. Proof. Let us proceed by induction on N =max(h(y1) ::: h(yn)). For N =1,i.e.,if yi 2 X , i =1 ::: n, the assertion is obvious. Let us prove this fact for an arbitrary N > 1 assuming that the assertion is already proved for the positiveintegers less than N . If max(h(y1 ) ::: h(yn )) = N , then, to be de nite, set h(yi )= N for i 6 k and h(yi ) k. In this case, any element yi , i 6 k, can be represented in the form yi = fi(yi1 ::: yisi ), where h(yij ) k,be neighborhoods of elements yj , j> k. By the induction assumption, there exist elements zij 2 Vij and zj 2 Vj such that the sets Xzij and Xzj are pairwise disjointand l(zij )=#Xzij and l(zj )=#Xzj . Then the elements zi = fi (zi1 ::: zisi ), i 6 k, and zk+1 ::: zn have the desired property.
1

4:3: Compatible Systems of Neighborhoods Let U 0 U be an F -subset of nite rank of a free AT -system A =(U F ). Toanypoint u 2 U 0 we assign a neighborhood Vu U of u. De nition. A system of neighborhoods fVu j u 2 U 0 g is said to be compatible with an F -subset 0 U if U 1) every neighborhood Vu is contained in some S -subset, 2) the neighborhoods Vu are pairwise disjoint, 3) if u = f (u1 ::: uk ), where ui 2 U 0 , then Vu = f (Vu ::: Vuk ). Note that, if Y = fy1 ::: yn g is a base of an F -subset U 0 , then any neighborhood system compatible with U 0 is uniquely de ned by some neighborhoods Vyi , i =1 ::: n, satisfying conditions 1) and 2). The following two assertions immediately result from the de nition of compatible systems.
1

Proposition 4.5. The union of neighborhoods entering any system compatible with a nite-rank F -subset U 0 U is an F -subset in U . Proposition 4.6. If f00Vu j u0 2 U 0 g is a system of neighborhoods compatible with an F -subset U000 , then, for any F -subset U U , the system of neighborhoods fVu j u 2 U 00 g is compatible with U . Theorem 4.1 (separation of F -subsets of nite rank). Any disjoint F -subsets U1 and U2 of e e nite rank in a free AT -system are contained in some disjoint open F -subsets U1 and U2 .
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neighborhoods fVu j u 2 U g compatible with U .Set
e U1 =
u2U

Proof. Introduce the F0 -subset U 0 = U _ U0 of nite rank and de ne an arbitrary system of
1 2

V
1

u

e U2 =

u2U

Vu:
2

e e It follows from Propositions 4.5 and 4.6 that U1 and U2 are disjointopen F -subsets containing U and U2 , respectively.

1

4:4: Topological Spaceof F -Subsets of a Free AT -System Denote by M = M X ] the set of all F -subsets of nite rank of a free AT -system A X ]= (U F ) with base X U . According to Subsection 4.1, the set M can be equipped with a Hausdor topology induced by the Hausdor topology on U . Namely, the base of neighborhoods of any F -subset U Y ] U with base Y = fy1 ::: yn g consists of the sets M (Vy ::: Vyn ) M , where Vyi ranges over a base of neighborhoods of the point yi for any i =1 ::: n. Let U0 = U y1 ::: yn ] be an arbitrary F -subset of nite rank of the free AT -system A X ] = (U F ), and let Vy ::: Vyn be neighborhoods of y1 ::: yn satisfying conditions 1) and 2) in Subsection 4.3. Theorem 4.2. For any F -subset U 0 U belonging to a neighborhood M (Vy ::: Vyn ) of an F -subset U0 , there is a natural isomorphism of the F -subsets, U 0 ! U0 . Proof. It follows from the de nition of the neighborhood M (V ::: Vyn ) that we have 0 = U (z1 ::: zn ), where zi 2 Vy , i = 1 ::: n. Let fVu j u 2 U0 g ybe the compatible system U i of neighborhoods generated by the neighborhoods Vyi . By the compatibility,every element z 2 U 0 belongs to one of the neighborhoods Vu . Since the neighborhoods Vu are pairwise disjoint, this relation de nes a mapping U 0 ! U0 taking zi to yi , i =1 ::: n, and preserving the multiplication. Since fy1 ::: yn g is a base set, it follows that fz1 ::: zn g is also a base set, and therefore the mapping U 0 ! U0 is an isomorphism of F -subsets. Corollary 4.1. The subsets Mn M of F -subsets of any chosen rank n are open and closed in M . Corollary 4.2. Any neighborhood of the form M (Vy ::: Vyn ) in the space of F -subsets M is homeomorphic to the Cartesian product of the corresponding neighborhoods, Vy Vyn .
1 1 1 1 1 1

4:5: Theorem on the Planar Subsets of a Free AT -System Theorem 4.3. If the base X of a free AT -system contains no isolated points, then the subset M 0 M of planar subsets is open and dense in M . Proof. Let M (Vy ::: Vyn ) be a neighborhood of an arbitrary F -subset U 0 = U y1 ::: yn] 2 U . By Proposition 4.4, there exist elements zi 2 Vyi , i =1 ::: n, such that the sets Xzi are pairwise disjointand l(zi)= #Xzi , i =1 ::: n. In this case, by Proposition 3.6, the F -subset U (z1 ::: zn ) contained in the neighborhood M (Vy ::: Vyn ) is planar. Hence, the family M 0 of planar subsets is dense in M . Let us prove that M 0 is open. Let U Y ], where Y = fy1 ::: yn g, be an arbitrary planar subset of U = U X ]. Introduce the F -subset U Z ] U Y ], where
1 1

Z=

n i
=1

X

yi

this set is planar because Z X . Choose an arbitrary compatible system of neighborhoods fVu j u 2 U Z ]g for U Z ]. Since U Y ] U Z ], it follows that the set U Y ] contains all elements of a compatible system of neighborhoods fVu j u 2 U Y ]g for U Y ].
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Consider the neighborhood M (Vy ::: Vyn ) of the planar subset U Y ]. Let us prove that all F -subsets in this neighborhood are planar. 0 0 Let U Y 0 ], where Y 0 = fy1 ::: yn g, be an arbitrary F -subset in this neighborhood, i.e., yi0 2 Vyi , i =1 ::: n. By Theorem 4.2, the elements yi0 form a base in U Y 0 ]. We claim that the F -subset U Y 0 ] is planar. Introduce the F -subset U Z 0 ] U Y 0 ], where
1

Z0 =

n i
=1

Xyi0 :

By Theorem 4.2, there is an isomorphism of F -subsets of the form : U Y 0 ] ! U Y ]. Let us extend this isomorphism to a mapping : U Z 0 ] ! U Z ]. By construction, every element u0 2 U Z 0 ] belongs to one of the neighborhoods Vu , u 2 U Z ]. Since these neighborhoods are pairwise disjoint, this de nes a mapping : U Z 0 ] ! U Z ] that coincides on U Y 0 ] with the original mapping. The extended mapping is surjective and preserves the operation of multiplication, the schemes, and the relation of subordination for the elements. However, is not a bijection in general. Suppose that the F -subset U Y 0 ] is not planar. Then the set U Y 0 ] is strictly contained in an F -subset U W ] U Z 0 ] with base W = fw1 ::: wr g, where r 6 n and

W=
Note that U = fu1 ::: ur g, where r 6 n and
n i=1

n i
=1

Wyi0 :

Uyi = U:

Wehave U W ] U Y ] and r( U W ]) 6 r. Since among the elements wi there exist elements that are strictly subordinated to at least one element yi0 and the mapping preserves the subordination relation, it follows that U W ] 6= U Y ]. This contradicts the condition that U Y ] is a planar subset. 4:6: Free Commutative and FreeIdempotent AT -Systems The topologization of free A-systems A X ]= (U F ) presented in Subsection 4.2 can also be extended to all free commutative and free idempotent A-systems. Thus, two new classes of topological A-systems arise, namely, the free commutative and free idempotent AT -systems. The above assertions and their proofs for free AT -systems, except for Proposition 4.5 and Theorem 4.1, remain valid for these new classes of AT -systems. In particular, the topology on these AT -systems is Hausdor . Proposition 4.5 and Theorem 4.1 hold for the free commutative AT -systems but are generally wrong for free idempotent AT -systems. The reason is that only expressions of the form u = f (u1 ::: un ), where there are at least two distinct elements of the form ui , enter the definition of compatible systems of neighborhoods. Therefore, sets of the form f (Vx ::: Vx) do not enter any compatible system of neighborhoods. Remark. The topologization presented here can be applied to an arbitrary tame A-system. However, in the general case, the resulting topology on U can be non-Hausdor , as can be seen from the following arguments. Let U = U X ] be a groupoid with base X . Suppose that elements x1 x2 2 X , x1 6= x2 , do not commute and any neighborhoods Vx and Vx of x1 and x2 , respectively, contain commuting elements y1 2 Vx and y2 2 Vx . Then the neighborhoods Vx x = Vx Vx and Vx x = Vx Vx of the elements x1 x2 and x2 x1 have nonemptyintersection. An example of such a groupoid can be constructed indeed.
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4:7: Free AT F -Systems Suppose that the structure of topological (Hausdor ) space is given not only on the base X U but also on the subsets Fn F of the n-ary operations of a free A-system A X ] = (U F ). Let us construct a topology on U induced by the topologies on X and Fn , n =1 2 :: : We shall now de ne bases of neighborhoods of the elements u 2 U by induction on their height h(u). For the elements of unit height, i.e., for the elements u 2 X , bases of neighborhoods are already (originally) de ned. If h(u)= n> 1, then the element u can uniquely be represented in the form u = f (u1 ::: un), where h(ui) < n, i = 1 ::: n. Let us de ne a base of neighborhoods of the element u as the family of sets Vf (Vu ::: Vun )= fu0 = f 0 (u01 ::: u0n ) j f 0 2 Vf u0i 2 Vui i =1 ::: ng where Vf and Vu ::: Vun range over bases of neighborhoods of the operation f 2 Fn and of the elements ui 2 U , respectively. It follows from the construction that the system A X ]= (U F ) equipped with the topology on U thus de ned is an AT F -system. We call it a free AT F -system. If the topology on F is discrete, then the AT F -topology coincides with the AT -topology. If the topology on F is not discrete, then the AT F -topology is weaker than the AT -topology, i.e., the identity mapping U ! U of the space U with AT -topology onto the space U equipped with the AT F -topology is continuous but not homeomorphic 3]. In the AT F -topology, the subsets of elements having an arbitrarily xed decomposition scheme are closed but not open. The subsets of elements whose decomposition schemes di er only on the labels at the vertices are open and closed. Note. One can also introduce the AT F -topology starting from the embedding U ! G X F ] of the set U in the free groupoid G X F ] generated by the set X F . Namely, G X F ] can be equipped with the AT -topology induced by the given topology on X F . The corresponding induced topology on the image of U coincides with the AT F topology on U . Consider the space of the nite-rank M X ]-subsets U 0 U of a free AT F -system A X ]= (U F ) with the topology induced by the topology on U . Proposition 4.7. The subspace M 0 X ] M X ] of planar subsets is dense in M X ]. Indeed, the AT F -topology on U is weaker than the AT -topology. Hence, the topology on M X ] induced by the AT F -topology on U is weaker than the topology induced on U by the AT -topology. Therefore, the assertion follows from the similar assertion for the AT -topology (Theorem 4.3). The above topologization of the free A-systems A X ] = (U F ) can also be used for the free commutative and free idempotent A-systems. Thus, two new classes of topological A-systems arise, namely, the free commutative and free idempotent AT F -systems. Proposition 4.7 remains valid for these systems as well.
1 1

4:8: Secondary Topology on the FreeIdempotent A-Systems Let A X ]= (U F ) be a free idempotent system whose base X is equipped with the structure of a Hausdor topological space. According to 4.6, the topology on X U induces a Hausdor topology on U .We refer to it as the primary topology on U . The following assertion immediately results from the de nition of this topology. Proposition 4.8. For any open subset V U and any f 2 F , the set f (V : : : V ) is also open. Introduce a new topology on U . For a base of neighborhoods we take the family of F -subsets 0 f = U (U 0 ) U generated by open subsets U 0 U with respect to the primary topology.We refer U to the topology thus obtained as the secondary topology on U . It follows from Proposition 4.8 that any subset open in the secondary topology is open in the primary topology as well however, the converse assertion fails. Thus, the secondary topology on U is weaker than the primary topology. As was shown above, the primary topology on U induced by a Hausdor topology on the base X is also Hausdor .
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Proof. Let x and y, x y 2 U , be arbitrary elements. We claim that, if x 6= y, then there exist neighborhoods Vx and Vy of the elements x y 2 U with respect to the primary topology on U such that the F -subsets Ux = U (Vx) and Uy = U (Vy ) in U generated by Vx and Vy , respectively, are disjoint. Denote by Z a nite subset formed0 by all elements z 2 X subordinated to x or y and by U 0 the F -subset of U generated by Z , i.e., U = U Z ]. Let fVu j u 2 U 0 g be an arbitrary system of neighborhoods (compatible with U 0 ) in the primary topology on U and let Uu = U (Vu ), u 2 U 0 ,be the F -subsets in U generated by the corresponding neighborhoods. Note that, since the neighborhood system is compatible, eachsubset Vu consists of elements of the same height. Therefore, this subset forms a base in Uu .We claim that Uu \ Uv = ? for any u v 2 U 0 , u 6= v . In particular, since x y 2 U 0 , this will imply the assertion of the theorem. Let us carry out the proof by induction on the height of the elements u and v .We assume rst that h(u) = h(v ) = 1, i.e., u v 2 Z , and thus Vu X and Vv X . Then it follows from the condition Vu \ Vv = ? that Uu \ Uv = ?. Let the assertion be already proved for the elements u and v of height less than n, and let maxfh(u) h(v )g = n, where n> 1. Suppose that the set W = Uu \ Uv is nonempty.Let w 2 W be an element of the minimal height. Then w belongs to one of the neighborhoods, to Vu or Vv . Indeed, otherwise the element w would be simultaneously representable in the forms w = f1 (u1 ::: um) where ui 2 Uu h(ui) h(v ), and therefore = h(u) = n and h(v ) < n. Since h(u) > 1, it follows that the element u can be represented in the form u = f (u1 ::: um) where ui 2 U 0 h(ui) 1

Theorem 4.4. If the topology on the base X topology on U is also Hausdor .

U is Hausdor , then the induced secondary

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De nition. An A-system A = (U F ) is said to be an AM -system if the support U of this system is equipped with the structure of a metric space such that the mapping (u1 ::: un ) 2 U n ! f (u1 ::: un ) 2 U is continuous for any n and any f 2 F . De nition. An A-system A = (U F ) is said to be an AM F -system if the sets U and F are equipped with the structures of metric spaces such that the mapping (f u1 ::: un) 2 Fn U n ! f (u1 ::: un ) 2 U is continuous for any n. If the topology on F is discrete, then these de nitions are equivalent. If an A-system A = (U F ) is an N -system, then every F -subset U 0 U admits a unique base. Using this fact, we can de ne, in terms of the metric (Archimedean or non-Archimedean) on the support U of the A-system, a metric (Archimedean or non-Archimedean, respectively) on the family Ln = Ln (U F ) of all F -subsets U 0 U of anygiven nite rank n. This metric on Ln can be introduced in several di erentways. If is an Archimedean metric on U , then we can set, for instance,
(U1 U2 ) = min
q
2

(x1 y

(1)

)+

+ 2 (xn y (n) )

for any F -subsets U1 and U2 with bases X = fx1 ::: xn g and Y = fy1 ::: yn g, respectively, where the minimum is taken over all permutations of the indices 1 ::: n. Another way to de ne a metric on Ln is (U1 U2)=min max(d(x1 y
(1)

) ::: d(xn y (n) )) :

(11)

Let us show that, if is an Archimedean or non-Archimedean metric on U , then formula (11) de nes an Archimedean or non-Archimedean metric Ln , respectively, i.e., for any F -subsets U1 , U2 , and U3 of rank n, in the Archimedean case, the triangle inequality (U1 U3 ) 6 (U1 U2)+ (U2 U3) holds, and in the non-Archimedean case, we have the stronger condition (U1 U3 ) 6 max( (U1 U3 ) (U1 U3)). Indeed, let X = fx1 ::: xn g, Y = fy1 ::: yn g, and Z = fz1 ::: zn g be bases on U1 , U2 , and U3 , respectively. It follows from the de nition of the metric on Ln that there are permutations 1 and 2 such that (U1 U2 )= max( (x1 y (1) ) ::: (xn y (n) )) (U2 U3 )= max( (y (1) z (1) ) ::: (y (n) z (n) )): Hence, (U1 U2 )+ (U2 U3 ) > max( (xi y (i) )+ (y (i) z (i) )) and max( (U1 U2 ) (U2 U3 )) > max( (xi y (i) ) (y (i) z (i) )): If the metric on U is Archimedean, then it follows from the rst inequality that (U1 U2 )+ (U2 U3 ) > (xi z (i) ) for any i = 1 ::: n, and therefore (U1 U2)+ (U2 U3 ) > (U1 U3). If the metric on U is nonArchimedean, then it follows from the other inequality that max( (U1 U2 ) (U2 U3 )) > (xi z (i) ) for any i, i =1 ::: n, and therefore max( (U1 U2 ) (U2 U3 )) > (U1 U3).
1 1 1 12 1 12 1 1 12 1 1 12 12 12

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5:2: Free AM -Systems with Archimedean and Non-Archimedean Metric Let A =(U F ) be a free A-system, and let the base X U of this system be equipped with an Archimedean or non-Archimedean metric (x y ). Let us construct an extension of this metric to the entire set U . We rst de ne the metric by induction on the height of elements on all S -subsets US U , i.e., on the subsets of elements with xed decomposition schemes S . On the S -subset of elements of unit height, i.e., on the subset X ,wehave the initial metric (which is already Archimedean or non-Archimedean). Suppose that the metric is already de ned on all S -subsets with elements of height less than k, and let US be an arbitrary S -subset with elements of height k. According to the de nition of an S -subset, one can nd an f 2 F and an S -subset US ::: USn with elements of height less than k, where n is the arityof f ,such that the S -subset US consists of the elements of the form
1

u = f (u1 ::: un)
i.e.,

h(ui ) 1

i

i =1 ::: n

US = US USn : By the induction assumption, the metric on the S -subsets USi is already de ned. If this metric is Archimedean, then we de ne (u v )for any elements u = f (u1 ::: un ) and v = f (v1 ::: vn ) in US by the formula p (u v )= 2 (u1 v1 )+ + 2 (un vn ): (12) Obviously, the function on US US satis es all axioms of Archimedean metric. Remark. Certainly, the way of de ning an Archimedean metric on US is not unique. For instance, a metric can also be de ned by the formula
(u v )= a1 (u1 v1 )+ + an (un vn ) where ai are arbitrarily chosen positivenumbers. For the case in which the metric on the S -subsets USi is non-Archimedean, let us de ne (u v ) for any elements u = f (u1 ::: un )and v = f (v1 ::: vn )in US by the formula (u v )= max( (u1 v1 ) ::: (un vn )): Obviously, the metric thus Let us extend the metric position scheme S ,wechoose Archimedean, then, for any u (13) de ned on US is non-Archimedean. de ned on the S -subsets in U to the entire set U . For any decoman arbitrary element yS 2 US . If the metric on the S -subsets US is 2 US and v 2 US with S1 6= S2 ,we set
1 2 1 2

(u v )= (u yS )+ (v yS )+ a where a is an arbitrarily chosen positivenumber. One can readily see that the metric thus de ned on U is Archimedean. Indeed, it su ces to verify the triangle inequality for any elements u 2 US , v 2 US ,and w 2 US in the following two cases: (1) S1 = S2 6= S3 , (2) S1 , S2 ,and S3 are pairwise distinct. In the rst case, wehave
1 2 3

(u w)= (u (v w)= (v (u v ) 6 (u

yS )+ (w yS )+ a yS )+ (w yS )+ a yS )+ (v yS )+ a:
1 3 2 3 1 2

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In the other case, the last inequality is replaced by the corresponding equality. In both cases, this implies the triangle inequality. If the metric on the S -subsets US is non-Archimedean, for any u 2 US and v 2 US , where S1 6= S2,we set (u v ) = max( (u yS ) (v yS ) a) where a is an arbitrarily xed positivenumber. Similarly to the Archimedean case, one can readily see that the metric thus de ned on U is non-Archimedean. The AM -systems thus de ned are said to be free AM -systems (Archimedean or non-Archimedean, respectively). Since the metric on the base X of a free AM -system induces the topology on X , to any free AM -system one can assign a free AT -system. The following assertion immediately results from the de nitions of free AT - and AM -systems. Proposition 5.1. The topology on the support U of a free AM -system A =(U F ) induced by an Archimedean or non-Archimedean metric on U coincides with the topology on U induced by the corresponding AT -system. An extension of an Archimedean or non-Archimedean metric with base X U to the support U can be de ned in a similar way for each free idempotent or free commutative A-system A =(U F ). The de nition can be extended to all free idempotent A-systems without modi cations. For a free commutative system, relations (12) and (13) in the de nition of the distance must be replaced by the relations q (u v ) = min 2 (u1 v (1) )+ + 2 (un v (n) ) and (u v ) = min(max( (u1 v1 ) ::: (un vn ))) respectively, where the minimum is taken over all permutations of the indices 1 ::: n. Thus, two new families of AM -systems with Archimedean and non-Archimedean metric arise, namely, free idempotent and free commutative AM -systems.
1 2 1 2

5:3: Secondary Non-Archimedean Metric on FreeIdempotent A-Systems Let A X ]= (U F ) be a free idempotent A-system, and let its base X U be equipped with a non-Archimedean metric d. Let us construct an extension of this metric to the support U of the A-system this extension di ers from that in Subsection 5.2. Let us de ne a metric d on S -subsets of U in the same way as in Subsection 5.2, i.e., by formula (13), using the induction on the height of elements. Let us extend the metric d to the entire set U , i.e., let us de ne d(u v )for any elements u v 2 U belonging to di erent S -subsets. To be de nite, let h(u) 6 h(v ). Then h(v ) > 1, and therefore the element v can uniquely be represented in the form v = f (v1 ::: vk ) where h(vi) RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 10 No. 1 2003


dean metric.

Proposition 5.2. The function d(u v) on U thus de ned satis es the axioms of non-Archime-

Proof. It su ces to show that 1) d(u v) 6= 0 if u and v belong to distinct S -subsets, and 2) d(u w) 6 max(d(u v ) d(v w)) if u, v , and w do not belong simultaneously to the same S -subset. Let us carry out the proof by induction on the height of elements. The assertion is trivial for the elements of unit height because these elements belong to the same S -subset. Suppose that the assertion is already proved for the elements of height less than n. Suppose that d(u v ) = 0 for some elements u v belonging to di erent S -subsets and assume that h(u) 6 h(v )= n. If h(u) d(xz (xy)z)= max(d(xz xy) d(xz z )):
Further, since the decomposition schemes of the elements xz and zy coincide and since h(xz ) >h(z ), it follows that

d(xz xy ) = max(d(x x) d(z y ))= d(z y) d(xz z ) = max(d(x z ) d(z z )) = d(x z):
Thus,

d(xz (xy)z ) = max(d(z y ) d(x z)):

Remark. In the de nition of the secondary metric, the assumption that the A-system in question is idempotent turns out to be substantial. Indeed, otherwise there exist elements of the form v = f (u : : : u), where h(u) < h(v ). By (14), for these elements, we have d(u v ) = 0, which contradicts the de nition of the metric d. Let us compare the secondary metric on U with the metric introduced in Subsection 5.2 (which we call primary metric ). It follows from the de nition of these metrics that they coincide on any
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S -subset US U with a xed decomposition scheme S .At the same time, the subset US is open in the topology on U induced by the primary metric but is not open with respect to the topology on U induced by the secondary metric. Thus, the secondary metric induces a topology on U which is weaker than the primary one. Let A X ] = (U F ) be a free idempotent system with secondary metric generated by a nonArchimedean metric d on the base X of the system. Denote by = r (u) the ball of radius r in U centered at the point u 2 U , i.e.,
r

(u)= fv 2 U j d(u v ) 6 rg:

Note that, in a non-Archimedean metric, anypoint of the ball is a center of this ball. Theorem 5.1. Let k be the least height of the elements belonging to the ball r (u), and let V r (u) be the subset of al l elements of height k in this bal l. Then
r

(u)= U (V )

e where V = U (V ) is the F -subset of U generated by V

U. Proof. Since for the center of the ball one can takeeachpoint u of this ball, we can assume that u 2 V . Then h(u)= k and V = fv 2 U j h(v )= k d(u v ) 6 rg: e e Let us rst prove that V . Denote by V (n) the subset of elements of height n in V = U (V ) (1) (1) (m) with respect to V . By de nition, V = V , and therefore V . Assume that V for any m < n, and let v 2 V (n) . Represent v in the form v = f (v1 ::: vl ), where h(vi ) < h(v ), i =1 ::: l. By the induction assumption, wehave vi 2 , and therefore d(u vi) 6 r, i =1 ::: l. Since d(u v ) = maxi (d(u vi)) by the de nition of the metric d, it follows that d(u v ) 6 r, and therefore v 2 . Conversely, let v 2 . Then h(v ) > k. If h(v ) = k, then v 2 V , and therefore v 2 U (V ). Assume that all elements v 2 of height less than n (where n > k) belong to U (V ), and let h(v ) = n. Let us represent v in the form v = f (v1 ::: vl ), where h(vi ) < h(v ), i = 1 ::: l. Since d(u v ) = maxi (d(u vi)), it follows from d(u v ) 6 r that d(u vi) 6 r, and therefore vi 2 , i =1 ::: l.Thus, vi 2 U (V ), i =1 ::: l,by the induction assumption. Then v 2 U (V ).
1. Kurosh, A. G., Lectures on General Algebra, M.: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1962 English transl. in: Lectures in General Algebra, Oxford: Pergamon, 1965. 2. Cohn, P. M., Universal Algebra, Dordrecht: D. Reidel, 1981. 3. Boruvka, O., Grund lagen der Gruppoid- und Gruppenteorie, Berlin: VEB Deutscher Verlag der Wissenschaften, 1960 English transl. in: Boruvka, O., Foundations of the Theory of Groupoids and Groups, New York: Halsted, 1976. 4. Birkho , G., Lattice Theory, 3rd ed., AMS Colloquium Publications, vol. XXV, 1996. 5. Stern, M., Semimodular Lattices. Theory and Applications, Cambridge: Cambridge University Press, 1999.

REFERENCES

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