Structural and universal completeness in algebra and logic

TL;DR

This paper explores algebraic and logical notions of completeness, providing new characterizations for various algebraic structures and applying these to substructural logics and MTL-algebras.

Contribution

It introduces novel algebraic characterizations of universal and structural completeness and applies these to specific varieties of lattices and residuated structures.

Findings

Passive structural completeness in substructural logics relates to classical contradictions being explosive.

Characterization of passively structurally complete varieties of MTL-algebras.

New algebraic criteria for universal completeness in quasivarieties.

Abstract

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.