In this note we give a presentation for the monoid IO n of all order-preserving transformations of a nchain whose ranges are intervals. We also consider the submonoid IO − n of IO n consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.

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In this paper we study several structural properties of the monoids POP2, of all injective orientation preserving partial transformations on a chain with n elements: we establish descriptions for the ideals and for the congruences of these monoids and we show that POPI, is a 2-generated semigroup, for all n E N.Our main aim is to give a presentation for these monoids.

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In this paper we calculate presentations for some natural monoids of transformations on a chain X First we consider n [ n ], the monoid of all full [partial] transformations on X n that preserve or reverse the order. Two other monoids of partial transformations on X n we look at are n and n -the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.