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<p>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, 3 of which are commutative. Among them, 4 are abelian, and the remaining non-abelian dimonoids form 2 pairs of dual dimonoids. Furthermore, there are exactly 5 pairwise non-isomorphic trivial dimonoids of order 2. For dimonoids of order 3, we prove that there are precisely 14 pairwise non-isomorphic commutative dimonoids, including 12 trivial dimonoids and a single pair of non-abelian non-trivial 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 non-commutative non-trivial ones. In addition, we demonstrate the existence of at least 26 pairwise non-isomorphic non-abelian non-commutative dimonoids of order 3. Among them, there are exactly 6 pairs of trivial dual dimonoids and at least 7 pairs of non-trivial dual dimonoids.</p>

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CLASSIFICATIONS OF DIMONOIDS WITH AT MOST THREE ELEMENTS

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • Volodymyr M. Gavrylkiv

摘要

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, 3 of which are commutative. Among them, 4 are abelian, and the remaining non-abelian dimonoids form 2 pairs of dual dimonoids. Furthermore, there are exactly 5 pairwise non-isomorphic trivial dimonoids of order 2. For dimonoids of order 3, we prove that there are precisely 14 pairwise non-isomorphic commutative dimonoids, including 12 trivial dimonoids and a single pair of non-abelian non-trivial 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 non-commutative non-trivial ones. In addition, we demonstrate the existence of at least 26 pairwise non-isomorphic non-abelian non-commutative dimonoids of order 3. Among them, there are exactly 6 pairs of trivial dual dimonoids and at least 7 pairs of non-trivial dual dimonoids.