Singly generated quasivarieties and residuated structures

Abstrakt

Familiar logics often have an algebraic counterpart that is a quasivariety K of algebras; in many cases it is a variety. In this situation, the derivable inference rules of the logic may or may not be determined by a single set of ‘truth tables’, i.e., by the operation tables of a single algebra A ∈ K. It turns out that if some member of K determines the finite rules of the logic, then another member determines all of the rules, so what is needed is only that K be generated by a single algebra—briefly, K = Q(A) for some A ∈ K. Even when K is a variety, it must be generated as a quasivariety by one of its members, if the generator is to determine rules (as opposed to theorems only).