Ranks and presentations of some normally ordered inverse semigroups

In this paper we compute the rank and exhibit a presentation for the monoids of all $P$-stable and $P$-order preserving partial permutations on a finite set $\mathrm{\Omega }$, with $P$ an ordered uniform partition of $\mathrm{\Omega }$. These (inverse) semigroups constitute a natural class of generators of the pseudovariety of inverse semigroups $\mathsf{N}\mathsf{O}$ of all normally ordered (finite) inverse semigroups.