## Abstract

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} ).

Original language | English |
---|---|

Pages (from-to) | 184-200 |

Number of pages | 17 |

Journal | Semigroup Forum |

Volume | 69 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 2004 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory