| Title:
|
Characterizing pure, cryptic and Clifford inverse semigroups (English) |
| Author:
|
Petrich, Mario |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
1099-1112 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
An inverse semigroup $S$ is pure if $e=e^2$, $a\in S$, $e<a$ implies $a^2=a$; it is cryptic if Green's relation $\mathcal {H}$ on $S$ is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-$\mathcal {V}$ varieties, for varieties $\mathcal {V}$ of inverse semigroups considered. (English) |
| Keyword:
|
inverse semigroup |
| Keyword:
|
pure inverse semigroup |
| Keyword:
|
cryptic inverse semigroup |
| Keyword:
|
Clifford semigroup |
| Keyword:
|
group-closed inverse semigroup |
| Keyword:
|
pure variety |
| Keyword:
|
completely semisimple inverse semigroup |
| Keyword:
|
combinatorial inverse semigroup |
| Keyword:
|
variety |
| MSC:
|
20M07 |
| MSC:
|
20M20 |
| idZBL:
|
Zbl 06433716 |
| idMR:
|
MR3304800 |
| DOI:
|
10.1007/s10587-014-0155-0 |
| . |
| Date available:
|
2015-02-09T17:43:13Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144163 |
| . |
| Reference:
|
[1] Mills, J. E.: Combinatorially factorizable inverse monoids.Semigroup Forum 59 (1999), 220-232. Zbl 0936.20054, MR 1847288, 10.1007/PL00006005 |
| Reference:
|
[2] Pastijn, F., Volkov, M. V.: Minimal noncryptic e-varieties of regular semigroups.J. Algebra 184 (1996), 881-896. Zbl 0862.20046, MR 1407875, 10.1006/jabr.1996.0289 |
| Reference:
|
[3] Petrich, M.: Inverse Semigroups.Pure and Applied Mathematics. A Wiley-Interscience Publication Wiley, New York (1984). Zbl 0546.20053, MR 0752899 |
| Reference:
|
[4] Reilly, N. R.: Minimal non-cryptic varieties of inverse semigroups.Q. J. Math., Oxf. II. Ser. 36 (1985), 467-487. Zbl 0582.20038, MR 0816487, 10.1093/qmath/36.4.467 |
| Reference:
|
[5] Sen, M. K., Yang, H. X., Guo, Y. Q.: A note on $\mathcal{H}$ relation on an inverse semigroup.J. Pure Math. 14 (1997), 1-3. MR 1658187 |
| . |
