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On the Classification of $n$-Valued Monoids and Groups of Order 3

Mikhail Kornev

TL;DR

This work classifies all $n$-valued monoids and groups of order $3$ by introducing a matrix-encoded representation of multiplication on a three-element set and solving the resulting quadratic associativity constraints, yielding six families of solutions and a complete isomorphism classification. It shows that all $n$-valued groups of order $3$ are commutative, identifies exactly two families of $n$-valued groups, and provides an explicit non-reversible family $\, ext{X}_n$; it also connects these results to the existing coset, involutive, and single-valued classifications, refining the landscape of $n$-valued algebraic structures. The paper thus clarifies the relationship between multivalued and single-valued group classifications, supplies explicit constructions, and situates the findings within the broader framework of $igstar$-involutive and coset groups. These contributions have implications for the structural theory of $n$-valued objects and their applications in mathematics and mathematical physics.

Abstract

The article presents a complete classification of $n$-valued monoids and groups of order 3. Important corollaries of this result are discussed.

On the Classification of $n$-Valued Monoids and Groups of Order 3

TL;DR

This work classifies all -valued monoids and groups of order by introducing a matrix-encoded representation of multiplication on a three-element set and solving the resulting quadratic associativity constraints, yielding six families of solutions and a complete isomorphism classification. It shows that all -valued groups of order are commutative, identifies exactly two families of -valued groups, and provides an explicit non-reversible family ; it also connects these results to the existing coset, involutive, and single-valued classifications, refining the landscape of -valued algebraic structures. The paper thus clarifies the relationship between multivalued and single-valued group classifications, supplies explicit constructions, and situates the findings within the broader framework of -involutive and coset groups. These contributions have implications for the structural theory of -valued objects and their applications in mathematics and mathematical physics.

Abstract

The article presents a complete classification of -valued monoids and groups of order 3. Important corollaries of this result are discussed.

Paper Structure

This paper contains 3 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Classification
  3. Corollaries

Key Result

Proposition 1

Let $X = \{e, x_1, x_2\}$ and Then every $n$-valued monoid on the set $X$ is represented by one of the following matrices $B = B_j$, provided that all fractional expressions yield non-negative integers: Moreover, two monoids corresponding to the series $B_i$ and $B_j$ ($i$ and $j$ not necessarily distinct) are isomorphic if and only if the matrix $B_i$ is proportional to the matrix $B_j$, or to

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Proposition
  • Corollary 1
  • Definition 3
  • Corollary 2
  • Corollary 3
  • Definition 4
  • Definition 5
  • Corollary 4
  • ...and 6 more