CONGRUENCES ON MONOIDS OF ORDER-PRESERVING OR ORDER-REVERSING TRANSFORMATIONS ON A FINITE CHAIN
Vítor Hugo Fernandes, Gracinda M. S. Gomes, Manuel M. Jesus · Glasgow Mathematical Journal · 2005
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.