Structural Monoids Associated to Equational Varieties

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