Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.24092 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915582336565248 |
|---|---|
| author | Gavrylkiv, Volodymyr M. |
| author_facet | Gavrylkiv, Volodymyr M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classifications of dimonoids with at most three elements Gavrylkiv, Volodymyr M. Group Theory Number Theory 18B40, 37L05, 22A15, 20D45, 20M15, 20B25 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. |
| title | Classifications of dimonoids with at most three elements |
| topic | Group Theory Number Theory 18B40, 37L05, 22A15, 20D45, 20M15, 20B25 |
| url | https://arxiv.org/abs/2510.24092 |