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We consider the monoid of order-preserving functions and order-reversing functions on an n -set. Over a field of characteristic not 2, we obtain a quiver presentation for the monoid algebra: two straight paths withn − 1and n vertices, respectively, and all compositions are zero. We also classify homomorphisms between induced left Schützenberger modules. We define a covering monoid with a distinction between order-preserving and order-reversing constant functions. We show that the algebra of the covering monoid is isomorphic to the direct product of two copies of the algebra of the monoid of all order-preserving functions on an n-set.

Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n -set