Abstract:

 Let PHn be the semigroups of order-decreasing and orderpreserving finite partial singular transformations on the natural order set Xn=｛1,2,3,…,n｝ if n≥3, and let P(n,r)=｛α∈PHn:|imα|≤r｝ be the two-sided star ideal of the semigroup PHn if 0≤r≤n-1. By analyzing the idempotent elements, the minimal idempotent generating set, rank and idempotent rank of the semigroup P(n,r) are characterized, respectively. Furthermore, it is shown that for 0≤l≤r, the relative rank of the semigroup P(n,r) with respect to itself each star ideal P(n,l).

Key words: order-decreasing; order-preserving; singular transformation semigroup; rank and idempotent rank; relative rank