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Reconstructing Classical Algebras via Ternary Operations

Jorge Fatelo, Nelson Martins-Ferreira

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.

Reconstructing Classical Algebras via 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.

Paper Structure

This paper contains 7 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boolean algebra
  4. De Morgan algebra
  5. Rings and near rings of characteristic two
  6. MV algebra
  7. conclusion

Key Result

Lemma 2.1

Let $(A,p,0,1)$ be a system consisting of a set $A$, together with a ternary operation $p$ and two constants $0,1\in A$ satisfying: Then, the following properties hold:

Theorems & Definitions (35)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 25 more