On sandwich sets and congruences on regular semigroups

Mario Petrich

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 1, page 27-46
  • ISSN: 0011-4642

Abstract

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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 e S ( e , f ) one-sided sandwich sets and characterize them abstractly where e , f ∈ E ( S ) . For a , a ' ∈ S such that a = a a ' a , a ' = a ' a a ' , we call S ( a ) = S ( a ' a , a a ' ) 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 ∈ S , we also define E ( a ) as the set of all idempotets e such that, for any congruence ρ on S , a ρ a 2 implies that a ρ e . We study the restrictions on S in order that S ( a ) or E ( a ) ∩ D a 2 be trivial. For ℱ ∈ { 𝒮 , ℰ } , we define ℱ on S by a ℱ b if F ( a ) ∩ F ( b ) ≠ ∅ . We establish for which S are 𝒮 or ℰ congruences.

How to cite

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Petrich, Mario. "On sandwich sets and congruences on regular semigroups." Czechoslovak Mathematical Journal 56.1 (2006): 27-46. <http://eudml.org/doc/31015>.

@article{Petrich2006,
abstract = {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.},
author = {Petrich, Mario},
journal = {Czechoslovak Mathematical Journal},
keywords = {regular semigroup; sandwich set; congruence; natural order; compatibility; completely regular element or semigroup; cryptogroup; regular semigroups; sandwich sets; congruences; natural order; compatibility; completely regular elements; cryptogroups},
language = {eng},
number = {1},
pages = {27-46},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On sandwich sets and congruences on regular semigroups},
url = {http://eudml.org/doc/31015},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Petrich, Mario
TI - On sandwich sets and congruences on regular semigroups
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 27
EP - 46
AB - 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.
LA - eng
KW - regular semigroup; sandwich set; congruence; natural order; compatibility; completely regular element or semigroup; cryptogroup; regular semigroups; sandwich sets; congruences; natural order; compatibility; completely regular elements; cryptogroups
UR - http://eudml.org/doc/31015
ER -

References

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