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

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 orderpreserving 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.