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# Article

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Keywords:
semigroup; normal cryptogroup; associate subgroup; representation; strong semilattice of semigroups; Rees matrix semigroup
Summary:
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.
References:
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[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
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