The quantale of order-preserving maps

TL;DR

This paper introduces two new composition operations for order-preserving maps that enable the set of such maps on a complete lattice to form quantale and co-quantale structures, expanding the algebraic framework of order theory.

Contribution

It defines two novel composition operations that extend the algebraic structures of order-preserving maps on complete lattices, unlike traditional composition.

Findings

The new operations coincide with usual composition for sup- and meet-preserving maps.

They enable $L^{L}$ to form quantale and co-quantale structures.

These structures are established when $L$ is a completely distributive lattice.

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.