Admissibility via Natural Dualities

TL;DR

This paper demonstrates how natural dualities can be used to characterize admissible clauses and quasi-identities in various algebraic structures, providing axiomatizations for several classes including lattices and algebras.

Contribution

It introduces a method to characterize admissibility in quasivarieties using natural dualities, offering explicit axiomatizations for multiple algebraic structures.

Findings

Axiomatizations for admissible clauses in bounded distributive lattices.

Characterizations for Stone, Kleene, and De Morgan algebras.

Application of duality theory to algebraic admissibility.

Abstract

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.