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Structural Monoids Associated to Equational Varieties proceedings of the american mathematical society Volume 117, Number 2, February 1993 STRUCTURALMONOIDS ASSOCIATEDTO EQUATIONALVARIETIES PATRICK DEHORNOY (Communicated by Andreas R. Blass) Abstract. Monoids of elementary transformations are associated to certain equational varieties in order to describe the corresponding free objects; in some regular cases, these monoids reduce to groups, and the word problem for the free objects of the variety is connected with the existence of normal forms for the members of the associated group. 1. The general framework Let "V be an equational variety [Co] and X be any set of generators. The free object of "V generated by X will be denoted by f ^(X). If (E) is a set of equations defining "V, and if M(X) denotes the free magma generated by X with respect to the signature of "V then fr(X) is the quotient structure of Af (X) under the least congruence =j that forces the equations in (E) to hold. We assume that an infinite sequence of variables, say Var, is fixed. A term built from Var and the operators (i.e., an element of M(Var)) will be called an abstract term, while the elements of M(L) are simply called terms. A pair of abstract terms (o, t) is said to be balanced if and only if the set of variables occurring (at least once) in a is exactly the set of variables occurring (at least once) in t . A presentation (E) is said to be balanced if all equations in (E) (considered as pairs of terms) are balanced. Notice that if (E) is a balanced set of equations then any identity that is a consequence of (E) is still balanced. We shall give a description of =^ in terms of some elementary transformations in the case that (E) is balanced. To get simpler notation, we shall assume in the sequel that the signature of 'V reduces to a single binary operator, say *; the general case could be treated in a similar way. If tp is any mapping from Var into Af (X), we still denote by cp the unique morphism of Af (Var) to Af (X) that extends tp, and write a9 for the image under cp of the abstract term a ; the set of all a9 's when cp ranges over all mappings of Var to M(L) is denoted by Substj(o-). Similar notation will occasionally be used for tp a mapping from Var to Af (Var), or from X to M(Z). Received by the editors February 6, 1990 and, in revised form, June 3, 1991. 1991 Mathematics Subject Classification. Primary 08B20. This work was supported in part by a CNRS grant PRC mathematiques et informatique. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 293 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 294 PATRICKDEHORNOY Definition. If (a , x) is a balanced pair of abstract terms and X is any (non- empty) set, p^'T is the partial mapping from Af (X) into itself whose domain is Substi(a) and that maps each term o9 to the corresponding term x9 . Notice that />£'T is functional because we assumed that (a, x) is balanced, and therefore each variable occurring in x already occurs in a, so that the value of a* completely determines the value of xf . Clearly the image of p^'x is Substz(T). It is obvious that if T is the image of 5 under some mapping p£•T with (a, x) an equation in (E), or more generally under the product of a finite number of such mappings, then S=^' T must hold. This, however, is not sufficient to obtain the whole congruence =£', since =%' has to identify not only the pairs (o'p, x9) (and (x9 , a9)) but also the pairs (Tx, T2) where T2 is obtained from Tx by replacing some subterm a9 of Tx by the corresponding term x9 . Therefore, we shall introduce for every equation (a, t) in (E) not only the two mappings p^r and p^'a but also a family of translated copies of these mappings indexed by the set of all possible positions where substitution can occur. Since we assumed that the signature contains only a single binary operator, terms can be viewed as binary trees in the usual way, so that the tree S * T is the binary tree whose left subtree is (the tree associated with) S and whose right subtree is (the tree associated with) T. As addresses for nodes in trees, we use finite sequences of O's and 1's, using 0 for the left direction. The set of all such sequences will be denoted by Seq, and the empty sequence (address of the root for all trees) by A. If 5 is a term and u is a member of Seq, we write S/u for the subterm (i.e., subtree) of S whose root is at position u, if such a subterm exists. The set of all addresses u such that S/u exists will be called the support of the term £. Now we make the following Definition, (i) For u in Seq and F a partial mapping from A/(X) into itself, we let tr„F be the ' w-translated' copy of F , i.e., the partial mapping G from Af (X) into itself such that • S is in Dom G if and only if S/u exists and is in Dom F; • in that case, G(S) is the unique term T such that 7)„ exists and is F(S/U), and T/v is equal to S/v for every v such that S/v exists and v is incompatible with u with respect to the prefix ordering on Seq, (ii) #£ is the monoid generated by all mappings trup^,T for u in Seq and (a, t) or (t , a) in (E), using composition. If we extend in the obvious way the usual definition of an action on a set to the case of partial transformations (we do not assume that the transformations are extended by the identity outside their domains), we easily get the following description of free members in the variety 'V . Lemma 1. Assume that (E) is a balanced set of equations for 'V and X is a nonempty set; then the monoid f)^ acts on M(L) and f^(X) is the correspond- ing homogeneous space. We shall see in Lemma 3 that f>^' is essentially the set of all identities that hold in y, and the point we wish to emphasize here is the fact that a natural monoid structure can be put on this set. This approach should be compared with the categorical approach that introduces groupoids and has led recently to License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRUCTURALMONOIDS ASSOCIATEDTO EQUATIONALVARIETIES 295 important developments in geometry and, in particular, in the domain of Yang- Baxter equations. (For an extended bibliography on this viewpoint, which seems to go back to Mac Lane, see [Ca].) It seems, however, that the present approach leads to different developments, such as the one introduced below concerning word problems. The aim of this note is not to present a complete study of the monoids r>^', but only to describe a few examples where they behave nicely. Before doing that, we will nevertheless establish that these monoids are rather intrinsic objects attached to the variety y, since they are largely independent of the choices of X and (E). We begin with two technical auxiliary results. Lemma 2. Let I. be a nonempty set; then the sets Substi(ff) and Substi(a') are equal if and only if there exists a permutation n o/Var such that a1 is an . Proof. Assume Substj;(cr) = Subst^er'). If the supports of a and a' are not equal then there exists u in Seq such that u is in the support of a but not in the support of a' (or conversely). Let tp be any mapping of Var into X: the support of a,<f is the support of a', so that u is not in the support of a,(f. But for any term S in Substj;(a), u is in the support of S, so o"f cannot be in Subsist). Hence the supports of a and a' must be equal. Now we construct a permutation n of Var such that a' is a11 by successively considering the extremal points u in the common support of a and a' (the addresses of the leaves in a and a') and setting n(o/u) = a',u . An obstruction in this process occurs when treating u only if one has, for some other v , either ay„ = a/v and a'iu ^ a'iv or °/u - a'iv ano- CT/»^ a/v • Assume the first case, and let S, T be distinct members of Af (X). Then define cp by cp(a/u) = S and cp(y) = T for all other variables y; the term a"p is in Substi(cr'), but it cannot be in Subst£(<7), since R/u = R/v holds for every R in Substx(ff), while a'9^ is (a'luY, that is S, and a'9jv is (o'^y, that is T. So we conclude that the construction of n succeeds. □ Lemma 3. For every nonempty f in r)^ , there exists a balanced pair of terms (Of, Xf) that is unique up to a permutation of the variables such that f is equal to p^'Xf ; moreover the equation (of,Xf) is derivable from (E).
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