The word problem for involutive residuated lattices and related structures
The paper presents results on the word problem for several varieties of residuated lattices. The crucial theorem proved in this paper states that the powerset monoid of a group can be embedded in a distributive involutive residuated lattice. Then the results on undecidability of the word problem for groups can be used in order to show that the word problem for the considered varieties is undecidable as well. More precisely, the result by Novikov and Boone, stating that there exists a finitely presented group with an undecidable word problem, implies that the word problem is undecidable for the following varieties: residuated lattices, involutive residuated lattices, any intersection of a nontrivial lattice variety and one of the previous varieties. The same is true also for the classes of finite members of each of the preceding varieties since the word problem for finite groups is also undecidable as was show by Slobodskoi. As a consequence of the previous results the quasi-equational theories for all mentioned varieties are undecidable.
scientific article; zbMATH DE number 5126579