arXiv subjects

Volodymyr M. Gavrylkiv

Publications and source records attributed to Volodymyr M. Gavrylkiv.

4 recordsLinked to original sources

arXiv2026

On the number of $g$-dimonoids of small order

We study $g$-dimonoids, that is, algebraic structures endowed with two associative binary operations satisfying a specified system of axioms. The paper investigates duality and isomor\-phisms of $g$-dimonoids and provides a complete characterization of all rectangular commuta\-tive $g$-dimonoids. Several new examples are constructed, including commutative iso-dual $g$-dimonoids that are not abelian, and noncommutative nonabelian rectangular $g$-dimonoids that are not dimonoids. A complete classification of all two-element $g$-dimonoids is obtained. Finally, we present the results of computer computations determining the numbers of all pairwise non\-isomorphic $g$-dimonoids of orders up to~$5$, as well as all pairwise nonisomorphic commutative, abelian, and rectangular $g$-dimonoids of orders up to~$6$, obtained using \texttt{GAP}, \texttt{Python}, and \texttt{C++}.

math.GR↗

arXiv2026

On dimonoids of small order

This paper is devoted to the study of dimonoids, i.e., algebraic structures equipped with two associative binary operations satisfying a given system of axioms. We introduce several new classes of dimonoids and determine their automorphism groups and halos. In particular, we construct examples of iso-dual nonabelian dimonoids, as well as nonabelian dimonoids with nonempty halo. Using these constructions, we obtain a complete classification, up to isomorphism, of all nonabelian noncommutative nontrivial dimonoids of order 3, thereby resolving the problem concerning this classification. Consequently, all three-element dimonoids are classified up to isomorphism, yielding exactly 52 pairwise nonisomorphic dimonoids of order 3. Finally, we present the results of computer computations determining the numbers of all pairwise nonisomorphic dimonoids of orders up to 5, as well as all pairwise nonisomorphic commutative, abelian, and rectangular dimonoids of orders up to 6, obtained using GAP, Python, and C++.

math.GR↗

arXiv2026

Note on the number of doppelsemigroups of small order

We study doppelsemigroups, i.e., algebraic structures equip\-ped with two associative binary operations satisfying a specified system of axioms. We investigate duality and isomorphisms of doppelsemigroups and examine the relationships between commutative, abelian, strong, and rectangular doppelsemigroups. Several examples are constructed, including nontrivial iso-opposite doppelsemigroups, noncommutative iso-dual doppelsemigroups, nonabelian iso-cross-dual doppelsemigroups, and nonstrong rectangular iso-opposite doppelsemigroups. Furthermore, we refine the complete classification of nonisomorphic doppelsemigroups of order~3. Finally, we present computer-assisted calculations yielding the numbers of all pairwise nonisomorphic doppelsemigroups and strong doppelsemigroups of orders up to~$5$, as well as all pairwise nonisomorphic commutative, abelian, and rectangular doppelsemigroups of orders up to~$6$, obtained using \texttt{GAP}, \texttt{Python}, and \texttt{C++}.

math.GR↗

arXiv2025

Classifications of dimonoids with at most three elements

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.

math.GR↗