Limit-preserving Function (order Theory) - Background and Motivation

Background and Motivation

In many specialized areas of order theory, one restricts to classes of partially ordered sets that are complete with respect to certain limit constructions. For example, in lattice theory, one is interested in orders where all finite non-empty sets have both a least upper bound and a greatest lower bound. In domain theory, on the other hand, one focuses on partially ordered sets in which every directed subset has a supremum. Complete lattices and orders with a least element (the "empty supremum") provide further examples.

In all these cases, limits play a central role for the theory, which is supported by their interpretations in the practical applications of the various disciplines. Now it is no surprise that one also is interested in specifying appropriate mappings between such orders. From an algebraic viewpoint, this means that one wants to find adequate notions of homomorphisms for the structures under consideration. As usually, this is achieved by considering those functions that are compatible with the constructions that are characteristic for the respective orders. For example, lattice homomorphisms are those functions that preserve non-empty finite suprema and infima, i.e. the image of a supremum/infimum of two elements is just the supremum/infimum of their images. In domain theory, one often deals with so-called Scott-continuous functions that preserve all directed suprema.

The background for the definitions and terminology given below is to be found in category theory, where limits (and co-limits) in a more general sense are considered. The categorical concept of limit-preserving and limit-reflecting functors is in complete harmony with order theory, since orders can be considered as small categories of a certain kind.