Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite commutative monoids and finite commutative non-monoids of a given integer \(n=p^\alpha q^\beta\), for every integers \(\alpha , \beta \ge 2\) and different primes p and q. In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order \(n=p^\alpha q^\beta,\) we give the minimal generating set. Moreover, we prove that there are exactly \((p^{\alpha }-2)(q^{\beta }-2)\) non-isomorphic commutative non-monoids of order \(p^\alpha q^\beta\). The identification of non-group semigroups for the integers \(p^{2\alpha }\) and \(2p^\alpha\) is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order \(p^2\) is presented.

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关于给定阶的有限非群半群的识别

为每个正整数识别有限的非组半群是重要的，因为这种半群的许多应用在诸如计算机科学，数学和有限机器之类的各种科学分支中都起作用。2014年，针对每个正整数，确定了有限的非交换半边形体作为此类半群的一种。在这里，我们尝试为每个整数\（\ alpha，\ beta \ ge 2 \）和给定整数\（n = p ^ \ alpha q ^ \ beta \）识别有限可交换单等式和有限可交换非单态。不同素数p和q。为了识别可交换的类半群，我们给出一类由给定阶数的2生成的类半群，对于阶\（n = p ^ \ alpha q ^ \ beta，\）的可交换非类群，我们给出最小的发电机组。而且，我们证明确实存在\（（p ^ {\ alpha} -2）（q ^ {\ beta} -2）\）阶\（p ^ \ alpha q ^ \ beta \）。实现了对整数\（p ^ {2 \ alpha} \）和\（2p ^ \ alpha \）的非群半群的标识。还指定了这些组的同构组。作为这项研究的结果，提出了阿贝尔群与阶\（p ^ 2 \）的交换半群之间的有趣区别。