Numerical representability of fuzzy total preorders

Abstract

We introduce the concept of a fuzzy total preorder. Then we analyze its numerical representability through a real-valued order-preserving function defined for each α-cut.

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]).…”

Decomposition and Arrow-Like Aggregation of Fuzzy Preferences

“…Furthermore, the fact of existence of different non-equivalent kinds of transitivity definitions and connectedness as well as completeness (see also [27][28][29][30]), tells us that the consideration in the fuzzy context of some kind of fuzzy total preorder is not unique. (Other non-equivalent definitions of transitivity have been introduced in this literature, see e.g., [27,28,31,32]).…”

Open Questions in Utility Theory

“…Following [2], we pay now attention to the possibility of defining comparisons or preferences that are fuzzy instead of crisp. Suppose that we consider different kinds of orderings on a nonempty set X.…”

“…To put an example, one may say that a fuzzy relation F on a universe U is reflexive if F (t, t) = 1 holds for every t ∈ U . However, definitions such as asymmetry, transitivity, etc., as well as certain operations such as intersections, unions, complements, etc., depend on the choice of a suitable triangular norm (see [2]) for details). In fact, there are equivalent definitions of such concepts in the crisp setting (working with classical sets, that is, non-fuzzy), that, when extended to the fuzzy approach, are no longer equivalent, and give rise to many possible different theories and approaches, depending on the definitions considered in each fuzzy context.…”