AI 中文总结
本文研究$g$-双半群的代数性质，完成二元$g$-双半群分类，用GAP等工具算出阶数≤5的非同构$g$-双半群数量及阶数≤6的各类非同构$g$-双半群数量。
AI 中文摘要
我们研究$g$-双半群，即配备两个满足特定公理系统的结合二元运算的代数结构。本文探讨$g$-双半群的对偶性与同构性，完整刻画所有矩形交换$g$-双半群；构造多个新例子，包括非阿贝尔的交换同对偶$g$-双半群、非双半群的非交换非阿贝尔矩形$g$-双半群；完成所有二元$g$-双半群的完整分类。最后，我们给出计算机计算结果：使用\texttt{GAP}、\texttt{Python}和\texttt{C++}确定了阶数不超过5的所有两两非同构$g$-双半群的数量，以及阶数不超过6的所有两两非同构交换、阿贝尔和矩形$g$-双半群的数量。
英文摘要
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++}.
