Residuated Lattices, Regular Languages, and Burnside Problem

In this talk we are going to explore an interesting connection between the famous Burnside problem for groups, regular languages, and residuated lattices. Let K be a finitely axiomatized class of residuated lattices. Recall that the usual way of proving decidability of universal theory for K is to establish that K has the finite embeddability property (FEP) [2, 3]. It turns out that the method of proving FEP for various classes of residuated structures used in [2, 3] is tightly connected to the generalized Myhill theorem [5] which characterizes regular languages as languages which are downward closed with respect to a dual well quasiorder compatible with concatenation of words. Due to this connection, it is possible to convert some decidability proofs for classes of residuated lattices into sufficient regularity conditions for classes of languages. For example, one can use the proof of FEP for the variety of integral residuated lattices in order to show an already known fact that the class of languages closed under the following rule (corresponding to the well-known weakening rule)