MATHEMATICA BOHEMICA, Vol. 134, No. 3, pp. 301-318, 2009

The rank of a commutative semigroup

Antonio M. Cegarra, Mario Petrich

Antonio M. Cegarra, Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, e-mail:; 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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