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arXiv · 2606.28908

Note on the number of doppelsemigroups of small order

Volodymyr M. Gavrylkiv
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Abstract

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++}.

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Volodymyr M. Gavrylkiv. 2026-06-27. Note on the number of doppelsemigroups of small order. https://arxiv.org/abs/2606.28908

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