Representations of Symmetric Implication Algebras as Multicubes

TL;DR

This paper demonstrates that symmetric implication algebras can be represented as subalgebras of products involving cubic implication algebras and Boolean algebras, providing a structural understanding of these algebras.

Contribution

It introduces locally symmetric implication algebras and shows how symmetric implication algebras are generated from cubic implication algebras and Boolean algebras.

Findings

Symmetric implication algebras are generated from cubic and Boolean algebras.

Representation of locally symmetric implication algebras as subalgebras of product algebras.

Every symmetric implication algebra is covered by a locally symmetric implication algebra.

Abstract

We show that the variety of symmetric implication algebras is generated from cubic implication algebras and Boolean algebras. We do this by developing the notion of a locally symmetric implication algebra that has properties similar to cubic implication algebras and provide a representation of these algebras as subalgebras of a product of a cubic implication algebra and an implication algebra. We then show that every symmetric implication algebra is covered by a locally symmetric implication algebra.