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Completely $0$ -simple semigroup with the basis property

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box-65892, Riyadh-11566, Saudi Arabia

E-mail Address: iaaldayel@imamu.edu.sa

E-mail Address: akhalaf59@gmail.com

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and

Ahmad Al Khalaf

https://orcid.org/0000-0002-5818-0003

Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, P.O. Box-65892, Riyadh-11566, Saudi Arabia

E-mail Address: ajalkalaf@imamu.edu.sa

Corresponding author.

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https://doi.org/10.1142/S1793557121501412Cited by:0 (Source: Crossref)

Abstract

A semigroup $S$ is said to have the Basis Property if for any subsemigroup $T$ of a semigroup $S$ , any two bases for $T$ have the same cardinality. The structure of completely $0$ -simple semigroup with the Basis Property is described. In particular, we proved that each completely $0$ -simple semigroup $S$ has the Basis Property if and only if $S$ satisfies one of the following conditions:

(1)	$S$ is produced from a completely simple semigroup with adjoint zero.
(2)	$S$ is an isomorphic to Rees’s semigroup $M^{0} (G; I, Λ; P)$ over a group $G^{0}$ with sandwich matrix $P$ such that $\| G \| = 1$ , $min {\| I \|, \| Λ \|} = 2$ , in addition $P$ has a zero in every row and column.

Communicated by A. R. Rajan

Keywords:

AMSC: 20M05, 03D40

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