A rank for right congruences on inverse semigroups

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Publication details

JournalBulletin of the Australian Mathematical Society
DatePublished - Aug 2007
Issue number1
Number of pages14
Pages (from-to)55-68
Original languageEnglish


The S-rank (where 'S' abbreviates 'sandwich') of a right congruence p on a semigroup S is the Cantor-Bendixson rank of rho in the lattice of right congruences RC of S with respect to a topology we call the finite type topology. If every rho is an element of RC possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup B-0(G, I) is ranked if and only if G is ranked and I is finite.

We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) congruent to E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic J-class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.

Our notion of rank arose from considering stability properties of the theory T-S of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that T-B is a theory of B-sets that is superstable but not totally transcendental.

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