Algebraic functions in Lukasiewicz implication algebras

摘要

In this article we study algebraic functions in {->, 1}-subreducts of MV-algebras, also known as Lukasiewicz implication algebras. A function is algebraic on an algebra A if it is definable by a conjunction of equations on A. We fully characterize algebraic functions on every Lukasiewicz 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 Lukasiewicz 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 for all there exists! Lambda p approximate to q within the variety generated by the 3-element chain.