| Title:
|
Normal cryptogroups with an associate subgroup (English) |
| Author:
|
Petrich, Mario |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
289-305 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
semigroup |
| Keyword:
|
normal cryptogroup |
| Keyword:
|
associate subgroup |
| Keyword:
|
representation |
| Keyword:
|
strong semilattice of semigroups |
| Keyword:
|
Rees matrix semigroup |
| MSC:
|
20M10 |
| MSC:
|
20M17 |
| idZBL:
|
Zbl 06236413 |
| idMR:
|
MR3073960 |
| DOI:
|
10.1007/s10587-013-0019-z |
| . |
| Date available:
|
2013-07-18T14:48:08Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143313 |
| . |
| Reference:
|
[1] Blyth, T. S., Martins, P. M.: On associate subgroups of regular semigroups.Commun. Algebra 25 (1997), 2147-2156. Zbl 0880.20048, MR 1451685, 10.1080/00927879708825979 |
| Reference:
|
[2] Martins, P. M., Petrich, M.: Unary semigroups with an associate subgroup.Commun. Algebra 36 (2008), 1999-2013. Zbl 1146.20041, MR 2418372, 10.1080/00927870801947306 |
| Reference:
|
[3] Petrich, M.: The existence of an associate subgroup in normal cryptogroups.Publ. Math. Debrecen 73 (2008), 281-298. Zbl 1181.20051, MR 2466374 |
| Reference:
|
[4] Petrich, M., Reilly, N. R.: Completely Regular Semigroups.Canadian Mathematical Society Series of Monographs and Advanced Texts 23 John Wiley & Sons, New York (1999). Zbl 0967.20034, MR 1684919 |
| . |
