The word problem for involutive residuated lattices and related structures

Abstract

It will be shown that the word problem is undecidable for involutive residuated lattices, for finite involutive residuated lattices and certain related structures like residuated lattices. The proof relies on the fact that the monoid reduct of a group can be embedded as a monoid into a distributive involutive residuated lattice. Thus, results about groups by P. S. Novikov and W. W. Boone and about finite groups by A. M. Slobodskoi can be used. Furthermore, for any non-trivial lattice variety \( \user1{\mathcal{V}} \), the word problem is undecidable for those involutive residuated lattices and finite involutive residuated lattices whose lattice reducts belong to \( \user1{\mathcal{V}} \). In particular, the word problem is undecidable for modular and distributive involutive residuated lattices.