On strict lower and upper sections of fuzzy orderings

Fuzzy strict preference and social choice

“…In this subsection, for a best understanding of our conditions and results, we assume that (see also [2,11,[18][19][20]): P (x, y) > 0) is interpreted as "x is strictly preferred to y with the degree P (x, y)"; (ii) R(x, y) R(y, x) (i.e., P (y, x) = 0) is interpreted as "x is preferred to y"; and (iii) R(x, y) = R(y, x) (i.e., P (x, y) = P (y, x) = 0) is interpreted as "x is equivalent to y with the degree I (x, y)". Thus, Definition 5 is seen as follows:…”

“…Eq. (11) and the definition of R imply R(x, y) = R(y, z) = 1. Thus * 3 (x, y, z) = R(y, x) ∧ R(z, y) = < 1 = * 2 (x, y, z).…”

“…(iii) Fono and Gwét [11,Lemma 2] show that if R is min-transitive, then P 2 is negative transitive if and only if R satisfies condition T. This result is a particular case of the third result of Corollary 8 when P = P 2 .…”

“…Definition 3 (See Banerjee [1], Dutta [9], Fodor and Roubens [10], Fono and Gwét [11], Richardson [17]). Let I and P be two fuzzy binary relations on A.…”

“…We have CO ⊂ SFO * ⊂ H * 1,2 ⊂ H * 1 ⊂ H * . We also note that the sets CO, SFO * and H * are traditional and usual (see [1,3,4,8,9,11,17,18,20]). …”