A Space of Fuzzy Orderings

Abstract

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,  x  G), and X is a (non empty) fuzzy subset of the character group ï£ (A) = Hom(A,{1,–1}) satisfying: 1. X is a fuzzy closed subset of ï£ (A). 2.  an element e  A such that ï³(e) = – 1  ï³ ïƒŽ X. 3. Xïž :={a  A\ ï³(a) = 1  ï³ ïƒŽ X} = 1. 4. If f and g are forms over A and if x  D( f  g) then  y  D( f ) and z  D(g) such that x  D<y, z >. Observe, by 3, that the element e  A whose existence is asserted by 2 is unique. Also, e  1 (since ï³(1) = 1  ï³ ïƒŽ X). Notice that for a  A, the set X(a):= {ï³ ïƒŽ Xï³(a) = 1} is clopen (i.e. both closed and open) in X. Moreover, ï³(a) = – 1  ï³(– a) = 1 holds for any ï³ ïƒŽ X (by 2).