Utility representation theorems for Debreu separable preorders

Representing preorders with injective monotones

“…Representation of non-total preorders by multiutilities came later and was remarkably developed in Evren and Ok (2011). Although strict monotones can be traced back to Peleg (1970) and Richter (1966), there continue to be advances in the field (Bosi et al, 2020a;Herden & Levin, 2012; Re ´bille ´2019). In fact, it was only recently in Minguzzi (2013) where strict monotone multi-utilities were introduced and later in Alcantud et al (2013Alcantud et al ( , 2016 where they were further studied.…”

Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness

“…It should be noted that if R is a crisp binary relation, then the above definitions of T-τ-upper semicontinuity and T-τ-upper semiclosedness slightly generalize the concepts of upper semicontinuity of type 1 and respectively upper semicontinuity of type 2 as defined in Herden and Levin [8] in the case of a preorder on a topological space (X, τ).…”

“…On the other hand, it should be noted that there is a large number of contributions to the representation of nontotal binary relations (in particular nontotal preorders) in the crisp case (see e.g., Chapter 5 in Bridges amd Mehta [7], Herden and Levin [8], Herden [9], Mehta [10], Bosi et al [11], Bosi and Mehta [12], and Bosi and Zuanon [13]). This is the case when there are elements x, y ∈ X such that neither x y nor y x.…”

Maximal elements of quasi upper semicontinuous preorders on compact spaces

“…Remark 2.5 It is easily seen that a preorder on (X, τ ) is quasi upper semicontinuous provided that it is either upper semicontinuous of type 1 or upper semicontinuous of type 2 (see Herden and Levin 2012) or else there exists an upper semicontinuous weak utility u for the strict part ≺ of (in particular, admits an upper semicontinuous order-preserving function).…”