Coherent orders

Undominated Maximals: General Definition and Characterizations

“…With the exception of quasi-coherence and T -regularity, no definition provided here is new. In point of fact, not even quasi-coherence is new as it was previously examined in [13]. Furthermore, a regularity notion somehow related to T -regularity was used in [14,15] (and sufficient conditions for the T -regularity of a relation are also implicit in Theorem 3.5 in [16]).…”

“…The topological closure of a subset S of X is denoted by cl S, while the topological interior of S is denoted by int S. We say that B is conditionally locally nonsatiated for τ if [x ∈ X and B a (x) = ∅] ⇒ x ∈ cl B a (x); locally quasi-nonsatiated for τ if B is quasi-nonsatiated and conditionally locally nonsatiated for τ; and locally nonsatiated for τ if B is nonsatiated and conditionally locally nonsatiated for τ (if x ∈ cl B a (x) for all x ∈ X). (Figure 1 in [13] conveniently summarizes some implications about the various notions of nonsatiation. Clearly, conditional local quasi-nonsatiation (is implied by local quasi-nonsatiation but) neither implies nor is implied by quasi-nonsatiation.…”

“…Many simple classes of quasi-coherent relations are considered in [13]: see Theorem 6, Theorem 9 (and Corollary 3), Theorem 10 (and Corollary 4) and Theorem 12 of the mentioned article. Corollary 4 in [13] serves as a lemma in the rest of the paper. Lemma 1 ([13], Corollary 4).…”