Abstract:

Let PCn be the semigroup of all order\|decreasing and or
derpreserving partial transformations on a finitechain ［n］. For an arb
itrary integer r such that 3≤r≤n-1, the rank and idempotent rank of the
 semigroup PC(n,r)={α∈PCn: 〖JB(|〗Im(α)〖JB)|〗≤r} were studied. We s
howed that PC(n,r) is generated by idempotents of rank r. Furthermore, we
obtained that the rank and idempotent rank of PC(n,r) are both equal  to ∑〖
DD(〗n〖〗k=r〖DD)〗〖JB((〗〖HL(1〗nk〖HL)〗〖JB))〗〖JB((〗〖HL(1〗k-1r-1
〖HL)〗〖JB))〗.

Key words: transformation semigroup, order\, decreasing, orderpreserving, idempotent rank, rank