Preordered sets valued in a GL-monoid

Let M=(L,*)M=(L,*) be a GL-monoid. An M-valued preordered set is an L-subset endowed with a reflexive and M-transitive L-relation, it is essentially a category enriched in a quantaloid generated by M. This paper presents a study of M-valued preordered sets with emphasis on symmetrization and the Cauchy completion. The main result states that symmetrization and the Cauchy completion of M-valued preordered sets commute up to a natural isomorphism.

► Some properties of GL-monoids are obtained.
► Reflexive and M-transitive L-relations on L-subsets are studied as M-valued preorders.
► Symmetrization and the Cauchy completion of M-valued preordered sets commute.