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Commutative cancellative semigroups and rational vector spaces

Antonio M. Cegarra;  Mario Petrich — 2006

Bollettino dell'Unione Matematica Italiana

Representing a commutative cancellative subarchimedean semigroup S as ℕ i ⁢ ( G , I ) , we consider Hom ⁢ ( S , ℚ ) and Hom  ⁢ ( G , ℚ ) , where Q is the additive group of rational numbers. These sets can be given the structure of rational vector spaces. Suitable isomorphic copies of these vector spaces are found by means of certain functions related to some mappings introduced by T. Tamura.

The rank of a commutative semigroup

Antonio M. Cegarra;  Mario Petrich — 2009

Mathematica Bohemica

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...

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