Many-valued partial orders in a closed monoidal category

Abstract

This paper introduces partially ordered Ω-monoidal relation systems on a category C endowed with a closed monoidal structure, and therewith explores a categorical theory of many-valued partial orders in C. For a fixed C-object Ω and a partially ordered Ω-monoidal relation system ϒ on C, the notion of a many-valued preordered (resp. many-valued partially ordered) C-object is formulated as a ϒ-preordered (resp. ϒ-partially ordered) C-object. Furthermore, the category of ϒ-preordered (resp. ϒ-partially ordered) C-objects is constructed and applied to quantale-valued preorders (resp. partial orders) on ordinary sets as well as on L-sets.