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MATHEMATICA BOHEMICA, Vol. 134, No. 3, pp. 301-318, 2009
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The rank of a commutative semigroup

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Antonio M. Cegarra, Mario Petrich

* Antonio M. Cegarra*, Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, e-mail: ` acegarra@ugr.es`; * Mario Petrich*, 21420 Bol, Brac, Croatia

**Abstract:** 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 $S_\alpha$, we prove that $\rank S$ 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 $\rank S$; 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.

**Keywords:** semigroup, commutative semigroup, independent subset, rank, separative semigroup, power cancellative semigroup, archimedean component

**Classification (MSC 2000):** 20M10

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