Classifications of dimonoids with at most three elements
Volodymyr M. Gavrylkiv
TL;DR
This work delivers a complete classification, up to isomorphism, of all dimonoids with at most three elements. It proves there are $8$ two-element dimonoids (with $4$ abelian and the rest partitioned into two dual nonabelian pairs), and enumerates $14$ pairwise nonisomorphic commutative three-element dimonoids (12 trivial plus a dual nontrivial pair) and $17$ abelian three-element dimonoids (12 trivial plus $5$ nontrivial noncommutative). It also establishes the existence of at least $26$ pairwise nonisomorphic nonabelian noncommutative three-element dimonoids, organized into dual-pair families, and provides the automorphism-group data for all classified cases. The results leverage doppelsemigroups, duality, and detailed semigroup analyses to produce explicit catalogs that support small-dimonoid structure studies and broader dialgebra applications.
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.