The rank of a commutative semigroup

Antonio M. Cegarra; Mario Petrich

Mathematica Bohemica (2009)

  • Volume: 134, Issue: 3, page 301-318
  • ISSN: 0862-7959

Abstract

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The concept of rank of a commutative cancellative semigroup is extended to all commutative semigroups S by defining 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 α , we prove that rank S is the supremum of ranks of various S α . Representing a commutative separative semigroup S as a semilattice of its (cancellative) archimedean components, the main result of the paper provides several characterizations of rank S ; in particular if 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.

How to cite

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Cegarra, Antonio M., and Petrich, Mario. "The rank of a commutative semigroup." Mathematica Bohemica 134.3 (2009): 301-318. <http://eudml.org/doc/38094>.

@article{Cegarra2009,
abstract = {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.},
author = {Cegarra, Antonio M., Petrich, Mario},
journal = {Mathematica Bohemica},
keywords = {semigroup; commutative semigroup; independent subset; rank; separative semigroup; power cancellative semigroup; archimedean component; independent subsets; ranks; separative semigroups; power cancellative semigroups; Archimedean components; commutative semigroups; semilattices},
language = {eng},
number = {3},
pages = {301-318},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The rank of a commutative semigroup},
url = {http://eudml.org/doc/38094},
volume = {134},
year = {2009},
}

TY - JOUR
AU - Cegarra, Antonio M.
AU - Petrich, Mario
TI - The rank of a commutative semigroup
JO - Mathematica Bohemica
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 134
IS - 3
SP - 301
EP - 318
AB - 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.
LA - eng
KW - semigroup; commutative semigroup; independent subset; rank; separative semigroup; power cancellative semigroup; archimedean component; independent subsets; ranks; separative semigroups; power cancellative semigroups; Archimedean components; commutative semigroups; semilattices
UR - http://eudml.org/doc/38094
ER -

References

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  1. Cegarra, A. M., Petrich, M., 10.1007/s10998-004-0521-z, Period. Math. Hung. 49 (2004), 35-44. (2004) Zbl1070.20068MR2106464DOI10.1007/s10998-004-0521-z
  2. Cegarra, A. M., Petrich, M., 10.1007/s10474-005-0179-x, Acta Math. Hung. 107 (2005), 71-75. (2005) Zbl1076.20049MR2148936DOI10.1007/s10474-005-0179-x
  3. Cegarra, A. M., Petrich, M., Commutative cancellative semigroups of low rank, Preprint. 
  4. Clifford, A. H., Preston, G. B., The Algebraic Theory of Semigroups, Vol I, Math. Surveys No. 7, Amer. Math. Soc., Providence (1961). (1961) Zbl0111.03403MR0132791
  5. Grillet, P. A., Commutative Semigroups, Kluwer, Dordrecht (2001). (2001) Zbl1040.20048MR2017849
  6. Hall, R. E., Commutative cancellative semigroups with two generators, Czech. Math. J. 21 (1971), 449-452. (1971) Zbl0244.20074MR0286920
  7. Hall, R. E., 10.2140/pjm.1972.41.379, Pacific J. Math. 41 (1972), 379-389. (1972) Zbl0252.20065MR0306369DOI10.2140/pjm.1972.41.379
  8. Howie, J. M., M. J. Marques Ribeiro, 10.1007/s10012-000-0231-2, Southeast Asian Bull. Math. 24 (2000), 231-237. (2000) Zbl0967.20030MR1810060DOI10.1007/s10012-000-0231-2
  9. Petrich, M., On the structure of a class of commutative semigroups, Czech. Math. J. 14 (1964), 147-153. (1964) Zbl0143.03403MR0166284
  10. Petrich, M., Introduction to Semigroups, Merrill, Columbus (1973). (1973) Zbl0321.20037MR0393206

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