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

Published online by Cambridge University Press:  27 July 2005

VÍTOR H. FERNANDES ,
GRACINDA M. S. GOMES  and
MANUEL M. JESUS
VÍTOR H. FERNANDES
Affiliation:
Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal e-mail: vhf@fct.unl.pt, mrj@fct.unl.pt
GRACINDA M. S. GOMES
Affiliation:
Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal e-mail: ggomes@cii.fc.ul.pt
MANUEL M. JESUS
Affiliation:
Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal e-mail: vhf@fct.unl.pt, mrj@fct.unl.pt
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Abstract

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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.

Keywords

20M2020M0520M17
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 47 , Issue 2 , May 2005 , pp. 413 - 424
Copyright
2005 Glasgow Mathematical Journal Trust