A Notion of Rank for Right Congruences on Semigroups
Victoria A. R. Gould · Communications in Algebra · 2005
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 ± bℰ S 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 T S of ± bℰ S , has the model theoretic property of being stable. Moreover, T S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.