Varieties and quasivarieties of lattices with complementation

TL;DR

This paper explores classes of complemented lattices with additional properties, providing new constructions and axiomatizations for varieties including those close to Boolean algebras.

Contribution

It introduces a semidirect product-like construction for generating finite subdirectly irreducible modular lattices with complementation and axiomatizes small varieties near Boolean algebras.

Findings

Constructed infinitely many finite subdirectly irreducible modular lattices with complementation.

Axiomatized small varieties covering Boolean algebras.

Identified subclasses satisfying specific quasi-identities and De Morgan laws.

Abstract

We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses satisfying additional conditions, such as the quasi-identity $({x'\wedge y\approx 0} \;\&\; {x\wedge y'\approx 0})$ $\Rightarrow x\approx y$, modularity, or De Morgan's laws. We present a construction resembling a semidirect product that yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying this quasi-identity. We axiomatize small varieties, each of which covers the variety of Boolean algebras, generated by certain small modular lattices with De Morgan complementation.