Projectivity in (bounded) integral residuated lattices

TL;DR

This paper investigates projective algebras within varieties of bounded commutative integral residuated lattices, revealing conditions under which finitely presented algebras are projective, and exploring implications for unification theory.

Contribution

It provides an algebraic analysis of projectivity in residuated lattices using ordinal sums, expanding understanding of projective finitely presented algebras in various subvarieties.

Findings

Many varieties of divisible commutative integral residuated lattices have all finitely presented algebras projective.

Finite Boolean algebras are projective in varieties with a Boolean retraction term.

Results connect projectivity with unification theory in algebraic logic.

Abstract

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras. Moreover, we study varieties with a Boolean retraction term, showing for instance that in a variety with a Boolean retraction term all finite Boolean algebras are projective. Finally, we connect our results with the theory of Unification.