الفهرس | Only 14 pages are availabe for public view |
Abstract This introductorychapterisconsideredasabackgroundfor the materialincludedinthethesis.Also,someconceptsrelated fuzzy softtopologicalspaceshaveinvestigated. 1.1 Latticetheoreticalfoundation The aimofthissectionistocollecttherelevantdefinitionsand results fromlatticeaboutastrictlytwo-sidedcommutativequan- tale andanorderreversinginvolution.Also,somepropertiesof lattice arepresented. In thisthesis,weusenotations,whichisstandardforthe ”fuzzy mathematics”usuallywithoutexplanation.Apartialorder on aset L is abinaryrelation ” ” whichisreflexive,transitive and antisymmetric.Aposet L = (L;) is asetequippedwith a partialorder[13]. Wesayasubset D of L is directedprovided it isnon-emptyandeveryfinitesubsetof D has anupperbound in D. Wesayasubset C of L is totallyorderedorachainifit 1 Introduction is non-emptyandallelementsof C are comparableunder ” ” (that is,either x y or y x for allelements x; y 2 L). Clearly, eachtotallyorderedisdirected. Definition 1.1.1. [17, 23, 53] Let L = (L;) be aposet. (1) L is calledalattice,if x _ y 2 L; x ^ y 2 L for all x; y 2 L; (2) L is calledacompletelattice,if WS 2 L; VS 2 L for all S L; In particular, WL = 1L and VL = 0L; where 1L and 0L are theleastelementandthegreatestelement,respectively; (3) L is calleddistributive,if x _ (y ^ z) =(x _ y) ^ (x _ z) and x ^ (y _ z) =(x ^ y) _ (x ^ z) for all x; y;z 2 L; (4) L is calledacompletedistributivelattice(resp.distributive lattice), if L is acomplete(resp.alattice)anddistributive lattice. L0L = L |