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A notion of rank for right congruences on semigroups

Gould, V. (2005) A notion of rank for right congruences on semigroups. Communications in Algebra, 33 (12). pp. 4631-4656. ISSN 1532-4125

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Abstract

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (∅, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± bES of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory TS of ± bES, has the model theoretic property of being stable. Moreover, TS is superstable if and only if S is weakly right Noetherian. In the present article, we show that TS satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.

Item Type: Article
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 12 Jun 2009 15:16
Last Modified: 12 Jun 2009 15:16
Published Version: http://dx.doi.org/10.1080/00927870500276650
Status: Published
Publisher: Taylor & Francis
Identification Number: 10.1080/00927870500276650
URI: http://eprints.whiterose.ac.uk/id/eprint/5911

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