Logics of variable inclusion

Abstract

A variety of algebras K is called strongly irregular whenever it satisfies an identity of the kind f(x, y) ≈ x, where f(x, y) is any term of the language in which x and y really occur. On the other hand, an identity ϕ ≈ ψ is said to be regular provided that exactly the same variables occur in ϕ and ψ. A variety is regular, when it is defined by regular identities. Examples of (strongly) irregular varieties abound in logic, since every variety with a lattice reduct is (strongly) irregular as witnessed by the term f(x, y) := x ∧ (y ∨ x). The algebraic study of regular varieties traces back to the pioneering work of P lonka [7], who introduced a classoperator Pl(·) nowadays called P lonka sums, and used it to prove that any regular variety K can be represented as P lonka sums of a suitable strongly irregular variety V, in symbols Pl(V) = K. In this case K is called the regularization of V.