Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

Let 𝒫  n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫  n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫  n | and |E(𝒫  n )|/|𝒫  n | are obtained. Similar results for 𝒫  n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.