Quotient-reflective and bireflective subcategories of the category of preordered sets (Q719745)

The authors recall the notions of ``closedness'' and ``strong closedness'' in set-based topological categories and give characterizations of closed and strongly closed subobjects of an object in the category of preordered sets. They show that these notions yield appropriate closure operators in the sense of Dikranjan and Giuli. They then investigate relationships between these closure operators and the well-known up- and down-closures. Finally, zero-dimensionality and pre-Hausdorffness of a preordered set \((B,R)\) is characterized by the condition that the preorder \(R\) is an equivalence relation on the set \(B\).