Topology on Ordered Sets: Topological Dioids

This chapter is devoted to the study of topological properties, first in general ordered sets, then in dioids (this will eventually lead to the concept of topological dioids) and to the solution of equations of the fixed-point type.

Various types of topologies may be introduced, depending on the nature of the ordered sets considered. The simplest cases correspond to a totally ordered set, or to a product of totally ordered sets (e.g. Rⁿ with the partial order induced by the usual order on R). The relevant topologies on such sets are extensions of usual topologies. We will concentrate here on the more general case of partially ordered sets (or “posets”). In relation to these sets, we introduce in Sect. 2 two basic topologies: the sup-topology and the inf-topology.

Then we show in Sect. 3 that the sup-topology may be interpreted in terms of limit sup of increasing sequences; and likewise that the inf-topology may be interpreted in terms of the limit inf of decreasing sequences. The notions of continuity and semi-continuity for functions on partially ordered sets are introduced in Sect. 4.

We then discuss the fixed-point theorem, first in the context of general ordered sets (Sect. 5), and next in the context of topological dioids, in view of solving linear equations of the fixed-point type. Section 7 is devoted to the concept of p-stable element in a dioid which guarantees the existence of a quasi-inverse, and which turns out to be useful in the solution of various types of equations, whether linear (Sect. 7.2) or nonlinear (Sect. 7.3).

Finally, Sect. 8 introduces and discusses the concepts of residuation and of generalized solutions.