Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach

Abstract

We introduce a new approach that deals, jointly and in a unified manner, with the topics of numerical (continuous) representability of total preorders and interval orders. This setting is based on the consideration of increasing scales and the systematic use of a particular kind of codomain, that has a key lattice theoretical structure of a completely distributive lattice and allows us to use a single function (taking values in that codomain) in order to represent both kinds of binary relations.

Numerical Representability of Ordered Topological Spaces with Compatible Algebraic Structure

“…It is also noticeable that algebraic techniques have also recently been introduced in the search for classical numerical representations of totally preordered structures, as well as some other kinds of orderings, in a unified way (see e.g. [10,11]). …”

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

“…At this point, we may think of other alternative representations in which the codomain of the functions used is no longer the real line R, but, instead, some other suitable set, say Z, also endowed with some ordering ad hoc. In this direction, in some papers in the specialized literature, other codomains have already been used, e.g., nonstandard numbers (see, e.g., [135]), sets of fuzzy numbers (see [136]), suitable subsets of the unit square or similar (see [120,137,138]), or lexicographic products and powers (see [139,140]). The idea is that the codomain be so that in order to represent, say, total preorders, interval orders or semiorders only one mapping from X into Z (see also [141]).…”

Reinterpreting a fuzzy subset by means of a Sincov's functional equation

“…the first three chapters of [9], or else [4]. There are also other alternative numerical representations for total preorders as well as for other particular kinds of orderings defined on a universe (see Definition 4.2. below, and [2,8,11]). As a matter of fact, some of those alternative representations lean on different kinds of fuzzy numbers (see [10,12]).…”