On the monoid of order-preserving transformations of a finite chain whose ranges are intervals

Abstract We give a presentation for the monoid $$\mathscr{I}\mathscr{O}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> of all order-preserving transformations of an n -chain whose ranges are intervals. We also consider the submonoid $$\mathscr{I}\mathscr{O}_n^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> of $$\mathscr{I}\mathscr{O}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> consisting of order-decreasing transformations, for which we determine the cardinality, the rank and a presentation.