Relations between Ranks of Certain Semigroups

Let S be a finite semigroup and let ? be an aquivalence relation on S such that ρ is a congruence. Then ρ is a subsemigroup of S, the direct product of semigroups, and S/ρ={xρ:x in S} is a semigroup with (xρ)(yρ)=(xy)ρ. In [2,3] Howie and Ribeiro study the small, lower, intermediate, upper and large rank of a finite semigroups S. In this paper, we investigate some relation among ranks of semigroups S, ρ and S/ρ.