| Title:
|
On sandwich sets and congruences on regular semigroups (English) |
| Author:
|
Petrich, Mario |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
27-46 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
regular semigroup |
| Keyword:
|
sandwich set |
| Keyword:
|
congruence |
| Keyword:
|
natural order |
| Keyword:
|
compatibility |
| Keyword:
|
completely regular element or semigroup |
| Keyword:
|
cryptogroup |
| MSC:
|
20M10 |
| MSC:
|
20M17 |
| idZBL:
|
Zbl 1157.20035 |
| idMR:
|
MR2206285 |
| . |
| Date available:
|
2009-09-24T11:31:22Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128052 |
| . |
| Reference:
|
[1] K. Auinger: Free objects in joins of strict inverse and completely simple semigroups.J. London Math. Soc. 45 (1992), 491–507. MR 1180258 |
| Reference:
|
[2] K. Auinger: The congruence lattice of a strict regular semigroup.J. Pure Appl. Algebra 81 (1992), 219–245. Zbl 0809.20054, MR 1179099, 10.1016/0022-4049(92)90058-N |
| Reference:
|
[3] T. S. Blyth and M. G. Gomes: On the compatibility of the natural order on a regular semigroup.Proc. Royal Soc. Edinburgh A94 (1983), 79–84. MR 0700501 |
| Reference:
|
[4] J. M. Howie: An introduction to semigroup theory.Academic Press, London, 1976. Zbl 0355.20056, MR 0466355 |
| Reference:
|
[5] G. Lallement: Congruences et équivalences de Green sur un demi-groupe régulier.C.R. Acad. Sci., Paris 262 (1966), 613–616. Zbl 0136.26603, MR 0207872 |
| Reference:
|
[6] K. S. S. Nambooripad: Structure of regular semigroups.Mem. Amer. Math. Soc. 224 (1979). Zbl 0457.20051, MR 0546362 |
| Reference:
|
[7] K. S. S. Nambooripad: The natural partial order on a regular semigroup.Proc. Edinburgh Math. Soc. 23 (1980), 249–260. Zbl 0459.20054, MR 0620922 |
| Reference:
|
[8] M. Petrich: Introduction to semigroups.Merrill, Columbus, 1973. Zbl 0321.20037, MR 0393206 |
| Reference:
|
[9] M. Petrich: Inverse semigroups.Wiley, New York, 1984. Zbl 0546.20053, MR 0752899 |
| Reference:
|
[10] P. G. Trotter: Congruence extensions in regular semigroups.J. Algebra 137 (1991), 166–179. Zbl 0714.20053, MR 1090216, 10.1016/0021-8693(91)90086-N |
| Reference:
|
[11] P. S. Venkatesan: Right (left) inverse semigroups.J. Algebra 31 (1974), 209–217. Zbl 0301.20058, MR 0346078, 10.1016/0021-8693(74)90064-7 |
| . |
