A Space of Fuzzy Orderings - Tawfiq, L.N.M. (Ibn AL-Haitham Journal For Pure and Applied Sciences, 2017)
In this paper the chain length of a space of fuzzy orderings is defined, and various properties of this invariant are proved. The structure theorem for spaces of finite chain length is proved. Spaces of Fuzzy Orderings Throughout X = (X,A) denoted a space of fuzzy orderings. That is, A is a fuzzy subgroup of abelian group G of exponent 2. (see [1] (i.e. x 2 = 1, ï€¢ x ïƒŽ G), and X is a (non empty) fuzzy subset of the character group ï£ (A) = Hom(A,{1,â€“1}) satisfying: 1. X is a fuzzy closed subset of ï£ (A). 2. ï€¤ an element e ïƒŽ A such that ï³(e) = â€“ 1 ï€¢ ï³ ïƒŽ X. 3. Xïž :={a ïƒŽ A\ ï³(a) = 1 ï€¢ ï³ ïƒŽ X} = 1. 4. If f and g are forms over A and if x ïƒŽ D( f ïƒ… g) then ï€¤ y ïƒŽ D( f ) and z ïƒŽ D(g) such that x ïƒŽ D. Observe, by 3, that the element e ïƒŽ A whose existence is asserted by 2 is unique. Also, e ï‚¹ 1 (since ï³(1) = 1 ï€¢ ï³ ïƒŽ X). Notice that for a ïƒŽ A, the set X(a):= {ï³ ïƒŽ Xï‚½ï³(a) = 1} is clopen (i.e. both closed and open) in X. Moreover, ï³(a) = â€“ 1 ïƒ› ï³(â€“ a) = 1 holds for any ï³ ïƒŽ X (by 2).