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

Abstract

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++}.