A refinement of ternary Boolean algebras

TL;DR

This paper introduces a new algebraic structure with a ternary operation to better understand ternary Boolean algebras, exploring various interpretations and their implications for algebraic subvarieties.

Contribution

It proposes a refined algebraic framework for ternary Boolean algebras using a non-commutative ternary operation and explores different interpretations leading to new subvarieties.

Findings

Boolean algebras characterized as a subvariety with Church's conditioned disjunction

Different interpretations yield distinct algebraic subvarieties

Illustrations include rings and near-rings of characteristic 2

Abstract

An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church's conditioned disjunction, Boolean algebras are characterized as a subvariety. Different interpretations for the ternary operation lead to distinct subvarieties. Rings and near-rings of characteristic 2 are used to illustrate the procedure.