Abundancy and Ranks of Injective Order-Decreasing Partial Catalan Monoids
This work analyzes the semigroup of isotone, order-decreasing, injective partial transformations of a finite chain and its ideals and Rees quotients. Students learn how the abundant/ample hierarchy (via starred Green's relations) classifies these transformation semigroups, why the domain-restricted family is right but not left abundant, and how the ranks of the Rees quotients are computed and shown to match those of the two-sided ideals through Catalan-type counting.
2501.00285
Studies the semigroup IC_n of all isotone (order-preserving) and order-decreasing injective partial transformations of a finite chain, together with the subsemigroup of such maps whose domain exclude…