| Title:
|
The rank of a commutative semigroup (English) |
| Author:
|
Cegarra, Antonio M. |
| Author:
|
Petrich, Mario |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
134 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
301-318 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The concept of rank of a commutative cancellative semigroup is extended to all commutative semigroups $S$ by defining $\mathop{\rm rank}S$ as the supremum of cardinalities of finite independent subsets of $S$. Representing such a semigroup $S$ as a semilattice $Y$ of (archimedean) components $S_\alpha $, we prove that $\mathop{\rm rank}S$ is the supremum of ranks of various $S_\alpha $. Representing a commutative separative semigroup $S$ as a semilattice of its (cancellative) archimedean components, the main result of the paper provides several characterizations of $\mathop{\rm rank}S$; in particular if $\mathop{\rm rank}S$ is finite. Subdirect products of a semilattice and a commutative cancellative semigroup are treated briefly. We give a classification of all commutative separative semigroups which admit a generating set of one or two elements, and compute their ranks. (English) |
| Keyword:
|
semigroup |
| Keyword:
|
commutative semigroup |
| Keyword:
|
independent subset |
| Keyword:
|
rank |
| Keyword:
|
separative semigroup |
| Keyword:
|
power cancellative semigroup |
| Keyword:
|
archimedean component |
| MSC:
|
20M05 |
| MSC:
|
20M10 |
| MSC:
|
20M14 |
| idZBL:
|
Zbl 1197.20051 |
| idMR:
|
MR2561308 |
| DOI:
|
10.21136/MB.2009.140663 |
| . |
| Date available:
|
2010-07-20T18:05:02Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140663 |
| . |
| Reference:
|
[1] Cegarra, A. M., Petrich, M.: Commutative cancellative semigroups of finite rank.Period. Math. Hung. 49 (2004), 35-44. Zbl 1070.20068, MR 2106464, 10.1007/s10998-004-0521-z |
| Reference:
|
[2] Cegarra, A. M., Petrich, M.: The rank of a commutative cancellative semigroup.Acta Math. Hung. 107 (2005), 71-75. Zbl 1076.20049, MR 2148936, 10.1007/s10474-005-0179-x |
| Reference:
|
[3] Cegarra, A. M., Petrich, M.: Commutative cancellative semigroups of low rank.Preprint. |
| Reference:
|
[4] Clifford, A. H., Preston, G. B.: The Algebraic Theory of Semigroups, Vol I.Math. Surveys No. 7, Amer. Math. Soc., Providence (1961). Zbl 0111.03403, MR 0132791 |
| Reference:
|
[5] Grillet, P. A.: Commutative Semigroups.Kluwer, Dordrecht (2001). Zbl 1040.20048, MR 2017849 |
| Reference:
|
[6] Hall, R. E.: Commutative cancellative semigroups with two generators.Czech. Math. J. 21 (1971), 449-452. Zbl 0244.20074, MR 0286920 |
| Reference:
|
[7] Hall, R. E.: The translational hull of an $N$-semigroup.Pacific J. Math. 41 (1972), 379-389. Zbl 0252.20065, MR 0306369, 10.2140/pjm.1972.41.379 |
| Reference:
|
[8] Howie, J. M., M. J. Marques Ribeiro: Rank properties in finite semigroups II: the small rank and the large rank.Southeast Asian Bull. Math. 24 (2000), 231-237. Zbl 0967.20030, MR 1810060, 10.1007/s10012-000-0231-2 |
| Reference:
|
[9] Petrich, M.: On the structure of a class of commutative semigroups.Czech. Math. J. 14 (1964), 147-153. Zbl 0143.03403, MR 0166284 |
| Reference:
|
[10] Petrich, M.: Introduction to Semigroups.Merrill, Columbus (1973). Zbl 0321.20037, MR 0393206 |
| . |
