A Classification of Two-dimensional Endo-commutative Algebras over F_2

TL;DR

This paper introduces and classifies all two-dimensional endo-commutative algebras over the field F_2, providing a comprehensive list of their multiplication tables and clarifying their relation to commutative algebras.

Contribution

It provides a complete classification of 2D endo-commutative algebras over F_2, including all isomorphism classes and their multiplication tables.

Findings

Complete list of 2D endo-commutative algebras over F_2

Clarification of the difference between commutativity and endo-commutativity

Enumeration of all algebra isomorphism types

Abstract

We introduce a new class of algebras called endo-commutative algebras in which the square mapping preserves multiplication, and provide a complete classification of endo-commutative algebras of dimension 2 over the field F_2 of two elements. We list all multiplication tables of the algebras up to isomorphism. This clarifies the difference between commutativity and endo-commutativity of algebras.