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Abstract. The full order-preserving
transformation semigroup OT(X) on a poset X has long been studied. In this paper, we study the semigroup
(OT(X, Y), θ) where X and Y are chains, OT(X, Y) is the set of
all order-preserving maps from X
into Y, θ  OT(Y, X) and the operation * is defined by α * β =
αθβ for all α, β OT(X, Y). We characterize when (OT(X, Y), θ) is regular, (OT(X, Y), θ)  OT(X) and (OT(X, Y), θ) OT(Y).
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