Reconstructing Classical Algebras via Ternary Operations

TL;DR

This paper explores how ternary operations can be used to define and unify various classical algebraic structures, revealing new insights into their interrelations and foundational properties.

Contribution

It introduces structures with two constants and a ternary operation that are isomorphic to key algebraic systems like Boolean algebras and rings of characteristic two.

Findings

Ternary operations can characterize Boolean and de Morgan algebras.

Structures with two constants and a ternary operation are isomorphic to MV-algebras.

The approach unifies diverse algebraic systems through ternary operations.

Abstract

Although algebraic structures are frequently analyzed using unary and binary operations, they can also be effectively defined and unified through ternary operations. In this context, we introduce structures that contain two constants and a ternary operation. We demonstrate that these structures are isomorphic to various significant algebraic systems, including Boolean algebras, de Morgan algebras, MV-algebras, and (near) rings of characteristic two. Our work highlights the versatility of ternary operations in describing and connecting diverse algebraic structures.