Classifications of dimonoids with at most three elements

TL;DR

This paper provides a complete classification of small dimonoids with up to three elements, detailing their isomorphism classes, commutativity, and duality properties, thereby advancing the structural understanding of these algebraic objects.

Contribution

The paper offers the first comprehensive classification of all two- and three-element dimonoids, including their isomorphism classes, commutative and abelian cases, and duality relationships.

Findings

8 two-element dimonoids, 3 commutative, 4 abelian, 2 dual pairs

14 nonisomorphic commutative three-element dimonoids, including 12 trivial

17 abelian three-element dimonoids, with 12 trivial and 5 nontrivial noncommutative

Abstract

In this paper, we present complete classifications, up to isomorphism, of all two-element dimonoids, all commutative three-element dimonoids, and all abelian three-element dimonoids. We show that, up to isomorphism, there exist exactly 8 two-element dimonoids, of which 3 are commutative. Among these, 4 are abelian, and the remaining nonabelian dimonoids form 2 pairs of dual dimonoids. Furthermore, there are exactly 5 pairwise nonisomorphic trivial dimonoids of order 2. For dimonoids of order 3, we prove that there are precisely 14 pairwise nonisomorphic commutative dimonoids, including 12 trivial dimonoids and a single pair of nonabelian nontrivial dual dimonoids. We also establish that, up to isomorphism, there are 17 abelian dimonoids of order 3, consisting of 12 trivial commutative dimonoids and 5 noncommutative nontrivial ones. In addition, we demonstrate the existence of at least 26 pairwise nonisomorphic nonabelian noncommutative dimonoids of order 3. Among them, there are exactly 6 pairs of trivial dual dimonoids and at least 7 pairs of nontrivial dual dimonoids.