关于小阶双幺半群
On dimonoids of small order
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中文总结 AI 辅助
研究双幺半群，引入新类并确定其自同构群和晕，构造特殊双幺半群例子，完成 3 阶非阿贝尔非交换非平凡双幺半群同构分类，给出 3 阶双幺半群完整分类，还呈现了阶数至多为 5 和 6 的相关双幺半群数量的计算机计算结果。
中文摘要 AI 辅助
本文致力于双幺半群的研究，双幺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。我们引入了几类新的双幺半群，并确定了它们的自同构群和晕。特别地，我们构造了同构对偶非阿贝尔双幺半群以及具有非空晕的非阿贝尔双幺半群的例子。利用这些构造，我们得到了 3 阶所有非阿贝尔非交换非平凡双幺半群的同构分类，从而解决了该分类问题。进而得到了 3 阶所有双幺半群的同构分类，共有 52 个两两不同构的 3 阶双幺半群。最后，我们给出了使用 GAP、Python 和 C++进行计算机计算的结果，确定了阶数至多为 5 的所有两两不同构双幺半群的数量，以及阶数至多为 6 的所有两两不同构的交换、阿贝尔和矩形双幺半群的数量。
英文摘要
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++.
