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arXiv · 2510.24092

Classifications of dimonoids with at most three elements

Volodymyr M. Gavrylkiv
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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.

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Volodymyr M. Gavrylkiv. 2025-10-28. Classifications of dimonoids with at most three elements. https://arxiv.org/abs/2510.24092

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