Congruences on Monoids of Order-Preserving or Order-Reversing Transformations on a Finite Chain

This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain Xₙ with n elements. Namely, we consider the monoids ₙ and ₙ of all full, respectively partial, transformations on Xₙ that preserve or reverse the order, as well as the submonoid ₙ of ₙ of all its order-preserving elements. The inverse monoid ₙ of all injective elements of ₙ is also considered. We show that in ₙ any congruence is a Rees congruence, but this may not happen in the monoids ₙ , ₙ and ₙ . However in all these cases the congruences form a chain.