On certain finite semigroups of order-decreasing transformations I

A. Laradji, A. Umar

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Abstract

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

Original languageEnglish
Pages (from-to)184-200
Number of pages17
JournalSemigroup Forum
Volume69
Issue number2
DOIs
Publication statusPublished - Sep 2004

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ASJC Scopus subject areas

  • Mathematics(all)

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