Combinatorial Results for Semigroups of Order-Preserving Full Transformations

Abstract

Let O_n be the semigroup of all order-preserving full transformations of a finite chain, say X_n = {1, 2, ..., n}, and for a given full transformation α: X_n → X_n let f(α) = |{x ∈ X_n: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → O_n: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → O_n: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(O_n), the set of idempotents of O_n.

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1. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

   A. Laradji & A. Umar

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1. A. Laradji
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2. A. Umar
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Correspondence to A. Laradji or A. Umar.

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Laradji, A., Umar, A. Combinatorial Results for Semigroups of Order-Preserving Full Transformations. Semigroup Forum 72, 51–62 (2006). https://doi.org/10.1007/s00233-005-0553-6

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• Received: 06 December 2004

• Published: 12 March 2006

• Issue Date: February 2006

• DOI: https://doi.org/10.1007/s00233-005-0553-6

Keywords

• Combinatorial Result
• Finite Chain
• Full Transformation