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It is known that in the lattice of all varieties of /-groups, there is a unique maximal proper subvariety. Based on this fact we give a simple algorithm for deciding the word problem in free /-groups. Some other applications are also given. The word problem for free /-groups was shown to be solvable by Holland and McCleary [4] by techniques appropriate to /-groups. We give here a brief universal algebra proof of this theorem which is not much use in studying further properties of /-groups but which generalizes to any finitely presented variety V for which there is some effectively given list of identities, none of which are satisfied by V but at least one of which holds in any proper subvariety or V. It was pointed out in Evans, Neff, and Mandelberg [1, p. 261], that the word problem is solvable for free algebras in any finitely presented equationally complete variety and what we have here is a mild generalization of this result. We recall that a lattice ordered group is a set G which is (i) a lattice under a partial order x < y, and (ii) a group under the operations xy, x_1, 1, with the order relation compatible with the group multiplication, i.e. x < y implies ax < ay, xa < ya. The class of all such algebras is a variety or equationally defined theory, as is easily seen if we express the lattice axioms as identities in terms of the join and meet, the group axioms as identities, and the compatibility properties by x(y A z) = xy A xz, (y A z)x = yx A zx, x(y V z) = xyV xz, (y V z)x = yx V zx. It is this class of equationally defined structures we have in mind when we talk about the variety of /-groups or the equational theory of /-groups. THEOREM. The word problem for free l-groups is solvable. PROOF. This decision problem is equivalent to that of deciding for any given identity u{x\, x-i, X3,... ) = u(xi, 2:2, £3, ■ ■ ■ ) in the theory of /-groups whether or not the identity is a consequence of the axioms and hence holds in all /-groups. There is an algorithm Mi which enumerates all identities, which hold in all /groups and only such identities. If u = v is an /-group identity which holds in all /-groups, it will eventually appear in this enumeration. Let L be the variety of all /-groups. In the lattice of all subvarieties of L, there is a largest proper subvariety N which is covered by L (Holland [3]). N is the variety Received by the editors July 2, 1985 and, in revised form, August 22, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 06F15; Secondary 08B20, 08A50. This research was supported in part by NSF Grant MCS 8203615. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

The word problem for free lattice-ordered groups (and some other free algebras)