On strict lower and upper sections of fuzzy orderings

Strict lower sections, strict upper sections and open order intervals (strict order intervals) are classical 9 notions associated to crisp orderings (crisp total preorders). In this paper, we extend these notions to fuzzy orderings. We determine all the fuzzy orderings (they contain all the crisp orderings) which make it possible 11 to compare two fuzzy strict lower sections and two fuzzy strict upper sections. We then deduce the properties (equality, inclusion and intersection) of fuzzy strict order intervals. In this way, we obtain fuzzy extensions 13 of well-known properties of crisp strict sections and crisp strict order intervals, and we display one example of their applications. 15 c © 2003 Published by Elsevier Science B.V.