Universal representation of fuzzy sets´ membership functions

A generalized membership function of a fuzzy set A in a fixed set Z={z} is established as a mapping m(z;A):Z/spl rarr/X into a mathematical system S=(X;R), which is structured by a set R={R/sub i/,i/spl epsiv/I} of polyadic relations R/sub i//spl sube/X/sup r(i)/. The mathematical system S=(X;R) being interpreted as a "quality measurement scale" (QMS), a value m(z/sub 0/,;A)/spl epsiv/ X may be treated as a measure for the quality "membership in the fuzzy set A". For the generalized membership function a universal form is found, namely the form of the universal membership function u(z;A):Z/spl rarr/)X\´, which maps the fixed set Z into mathematical system S\´=(X\´;R/sub /spl ges//), the mathematical system S\´ being a universal representation for the initial system S=(X;R). The structure R/sub /spl ges// of the universal mathematical system is formed from so-called chain-dominance r(i)-adic relations R/sub /spl ges///sup r(i)/(R/sub 2//sup i/), which are induced by corresponding order relations R/sub 2//sup i/. Measurement theoretical interpretation for the universal representation S\´ of an arbitrary mathematical system S is given and the fundamental role of ordinal measurement scales in fuzzy sets theory is discussed.