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Volume 47 - Issue 2 - May 2005

Glasgow Mathematical Journal Volume 47 - Issue 2 -
Contents

Page 2 of 2

Research Article

  • A PARTITION RESULT WHICH IMPLIES FINE'S IDENTITY

  • WEI–YUEH CHEN, SEN–SHAN HUANG
  • Published online by Cambridge University Press:
    27 July 2005, pp. 405-411
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  • In this article, we derive a partition result and employ its special case to offer a new proof for an identity of N. J. Fine.

  • CONGRUENCES ON MONOIDS OF ORDER-PRESERVING OR ORDER-REVERSING TRANSFORMATIONS ON A FINITE CHAIN

  • VÍTOR H. FERNANDES, GRACINDA M. S. GOMES, MANUEL M. JESUS
  • Published online by Cambridge University Press:
    27 July 2005, pp. 413-424
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  • This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain $X_n$ with $n$ elements. Namely, we consider the monoids $\od_n$ and $\mpod_n$ of all full, respectively partial, transformations on $X_n$ that preserve or reverse the order, as well as the submonoid $\po_n$ of $\mpod_n$ of all its order-preserving elements. The inverse monoid $\podi_n$ of all injective elements of $\mpod_n$ is also considered.

    We show that in $\po_n$ any congruence is a Rees congruence, but this may not happen in the monoids $\od_n$, $\podi_n$ and $\mpod_n$. However in all these cases the congruences form a chain.

Page 2 of 2