The quantale of order-preserving maps

Abstract: In this paper, two new composition operations are defined among the order-preserving maps. They can act on order-preserving maps like the usual composition operation. They are coincide with the usual composition operation when the order-preserving maps are sup-preserving maps or meet-preserving maps. The usual composition operation can't endow the set $L^{L}$ of order preserving maps on a complete lattices $L$ with quantale or co-quantale structures. Luckily, two new operations can endow $L^{L}$ with a quantale and a co-quantale structures respectively, whenever $L$ is a completely distributive lattice.