Geodesic
On sandwich sets and congruences on regular semigroups
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 27-46 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $S$ be a regular semigroup and $E(S)$ be the set of its idempotents. We call the sets $S(e,f)f$ and $eS(e,f)$ one-sided sandwich sets and characterize them abstractly where $e,f \in E(S)$. For $a, a^{\prime } \in S$ such that $a=aa^{\prime }a$, $a^{\prime }=a^{\prime }aa^{\prime }$, we call $S(a)=S(a^{\prime }a, aa^{\prime })$ the sandwich set of $a$. We characterize regular semigroups $S$ in which all $S(e,f)$ (or all $S(a))$ are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every $a \in S$, we also define $E(a)$ as the set of all idempotets $e$ such that, for any congruence $\rho $ on $S$, $a \rho a^2$ implies that $a \rho e$. We study the restrictions on $S$ in order that $S(a)$ or $E(a)\cap D_{a^2}$ be trivial. For $\mathcal F \in \lbrace \mathcal S, \mathcal E\rbrace $, we define $\mathcal F$ on $S$ by $a \mathrel {\mathcal F}b$ if $F(a) \cap F (b)\ne \emptyset $. We establish for which $S$ are $\mathcal S$ or $\mathcal E$ congruences.
Let $S$ be a regular semigroup and $E(S)$ be the set of its idempotents. We call the sets $S(e,f)f$ and $eS(e,f)$ one-sided sandwich sets and characterize them abstractly where $e,f \in E(S)$. For $a, a^{\prime } \in S$ such that $a=aa^{\prime }a$, $a^{\prime }=a^{\prime }aa^{\prime }$, we call $S(a)=S(a^{\prime }a, aa^{\prime })$ the sandwich set of $a$. We characterize regular semigroups $S$ in which all $S(e,f)$ (or all $S(a))$ are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every $a \in S$, we also define $E(a)$ as the set of all idempotets $e$ such that, for any congruence $\rho $ on $S$, $a \rho a^2$ implies that $a \rho e$. We study the restrictions on $S$ in order that $S(a)$ or $E(a)\cap D_{a^2}$ be trivial. For $\mathcal F \in \lbrace \mathcal S, \mathcal E\rbrace $, we define $\mathcal F$ on $S$ by $a \mathrel {\mathcal F}b$ if $F(a) \cap F (b)\ne \emptyset $. We establish for which $S$ are $\mathcal S$ or $\mathcal E$ congruences.
Classification : 20M10, 20M17
Keywords: regular semigroup; sandwich set; congruence; natural order; compatibility; completely regular element or semigroup; cryptogroup 
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     author = {Petrich, Mario},
     title = {On sandwich sets and congruences on regular semigroups},
     journal = {Czechoslovak Mathematical Journal},
     pages = {27--46},
     year = {2006},
     volume = {56},
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     mrnumber = {2206285},
     zbl = {1157.20035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a3/}
}
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Petrich, Mario. On sandwich sets and congruences on regular semigroups. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 27-46. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a3/

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