Continuous Order-preserving Functions Research Articles

Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

We investigate properties of strongly useful topologies, i.e., topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.

Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

We present some lifting theorems for continuous order-preserving functions on locally and σ-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and σ-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions.

Continuous Order-Preserving Functions for All Kind of Preorders

Continuous Order-Preserving Functions for All Kind of Preorders

Preference orders and continuous representations

Abstract In this paper we prove some general theorems on the existence of continuous order-preserving functions on topological spaces with a continuous preorder. We use the concepts of network and netweight to prove new continuous representation theorems and we establish our main results for topological spaces that are countable unions of subspaces. Some results in the literature on path-connected, locally connected and separably connected spaces are shown to be consequences of the general theorems proved in the paper. Finally, we prove a continuous representation theorem for hereditarily separable spaces.

Constructive utility functions on Banach spaces

In this paper we prove the existence of continuous order-preserving functions on subsets of ordered Banach spaces using a constructive approach.

Continuous representability of homothetic preorders by means of sublinear order-preserving functions

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

Separation axioms in topological preordered spaces and the existence of continuous order-preserving functions

We characterize the existence of a real continuous order-preserving function on a topological preordered space, under the hypotheses that the topological space is normal and the preorder satisfies a strong continuity assumption, called IC-continuity. Under the same continuity assumption concerning the preorder, we present a sufficient condition for the existence of a continuous order-preserving function in case that the topological space is completely regular.

On lattices of continuous order-preserving functions

On lattices of continuous order-preserving functions

Coproducts of De Morgan algebras

The dual of the category of De Morgan algebras is described in terms of compact totally ordered-disconnected ordered topological spaces which possess an involutorial homeomorphism that is also a dual order-isomorphism. This description is used to study the coproduct of an arbitrary collection of De Morgan algebras and also to represent the coproduct of two De Morgan algebras in terms of the continuous order-preserving functions from the Priestley space of one algebra to the other algebra, endowed with the discrete topology. In addition, it is proved that the coproduct of a family of Kleene algebras in the category of De Morgan algebras is the same as the coproduct in the subcategory of Kleene algebras if and only if at most one of the algebras is not boolean.

Ordering Uniform Completions of Partially Ordered Sets

Let (P, ) be a (nearly) uniform ordered space. Let (P, ) be the uniform completion of (P, ) at . Several partial orders for P are introduced and discussed. One of these orders provides an adjoint to the functor which embeds the category of uniformly complete uniform ordered spaces in the category of uniform ordered spaces, both categories with uniformly continuous order-preserving functions. When P is a join-semilattice with uniformly continuous join, two of these orders coalesce to the unique partial order with respect to which P is a join-semilattice, P is a join-subsemilattice of P, and the join on P is continuous.