Algebraic functions in Łukasiewicz implication algebras

Abstract

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.