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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...

The structure of idempotent residuated chains

Wei Chen, Xian Zhong Zhao (2009)

Czechoslovak Mathematical Journal

In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation 𝒟 on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain...

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