The Undecidability of the Word Problem for Distributive Residuated Lattices

Abstract

Let A = (X|R) be a finitely presented algebra in a variety V. The algebra A is said to have an undecidable word problem if there is no algorithm that decides whether or not any two given words in the absolutely free term algebra T^v(X) represent the same element of A. If V contains such an algebra A, we say that it has an undecidable word problem. (It is well known that the word problem for the varieties of semigroups, groups and l-groups is undecidable.)

The main result of this paper is the undecidability of the word problem for a range of varieties including the variety of distributive residuated lattices and the variety of commutative distributive ones. The result for a subrange, including the latter variety, is a consequence of a theorem by Urquhart [7]. The proof here is based on the undecidability of the word problem for the variety of semigroups and makes use of the concept of an n-frame, introduced by von Neumann. The methods in the proof extend ideas used by Lipshitz and Urquhart to establish undecidability results for the varieties of modular lattices and distributive latticeordered semigroups, respectively.