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semigroup; normal cryptogroup; associate subgroup; representation; strong semilattice of semigroups; Rees matrix semigroup
Let $S$ be a semigroup. For $a,x\in S$ such that $a=axa$, we say that $x$ is an associate of $a$. A subgroup $G$ of $S$ which contains exactly one associate of each element of $S$ is called an associate subgroup of $S$. It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms. A normal cryptogroup $S$ is a completely regular semigroup whose $\mathcal H$-relation is a congruence and $S/\mathcal H$ is a normal band. Using the representation of $S$ as a strong semilattice of Rees matrix semigroups, in a previous communication we characterized those that have an associate subgroup. In this paper, we use that result to find three more representations of this semigroup. The main one has a form akin to the one of semigroups in which the identity element of the associate subgroup is medial.
[1] Blyth, T. S., Martins, P. M.: On associate subgroups of regular semigroups. Commun. Algebra 25 (1997), 2147-2156. DOI 10.1080/00927879708825979 | MR 1451685 | Zbl 0880.20048
[2] Martins, P. M., Petrich, M.: Unary semigroups with an associate subgroup. Commun. Algebra 36 (2008), 1999-2013. DOI 10.1080/00927870801947306 | MR 2418372 | Zbl 1146.20041
[3] Petrich, M.: The existence of an associate subgroup in normal cryptogroups. Publ. Math. Debrecen 73 (2008), 281-298. MR 2466374 | Zbl 1181.20051
[4] Petrich, M., Reilly, N. R.: Completely Regular Semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts 23 John Wiley & Sons, New York (1999). MR 1684919 | Zbl 0967.20034
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