A notion of rank for right congruences on semigroups

We introduce a new notion of rank for a semigroup S . The rank is associated with pairs (I ,rho), where rho is a right congruence and I is a rho-saturated right ideal. We allow I to be the empty set; in this case the rank of (empty set, rho) is the Cantor-Bendixson rank of rho 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 xi(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 xi(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.