On certain finite semigroups of order-decreasing transformations I

Let D_n (script O sign_n) be the semigroup of all finite order-decreasing (order-preserving) full transformations of an n-element chain, and let D(n,r) = {α ∈ D_n : |Im α| ≤ r} (C(n,r) = D(n,r) ∩ script O sign_n) be the two-sided ideal of D_n (D_n ∩ script O sign_n). Then it is shown that for r ≥ 2, the Rees quotient semigroup DP_r(n) = D(n,r)/D(n,r-1) (CP_r(n) = C(n,r)/C(n,r- 1)) is an R-trivial ( J-trivial) idempotent-generated 0*-bisimple primitive abundant semigroup. The order of CP_r(n) is shown to be 1 + (^n-1 _r-1) ( ⁿ_r ) /(n - r + 1). Finally, the rank and idempotent ranks of CP_r(n) (r < n) are both shown to be equal to ( ^n-1_r-1 ).