The rank of a commutative semigroup. (English) Zbl 1197.20051

Summary: The concept of rank of a commutative cancellative semigroup is extended to all commutative semigroups \(S\) by defining \(\text{rank\,}S\) 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 \(\text{rank\,}S\) 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 \(\text{rank\,}S\); in particular if \(\text{rank\,}S\) 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.


20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
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