A generalized membership function of a fuzzy set A in a fixed set Z = {z} is established as a mapping m(z;A): Z → X into a mathematical system S = (X; R), which is structured by a set R = {R_i, i qq I} of polyadic relations R_i qq X^r(i). The mathematical system S = (X; R) being interpreted as a 'quality measurement scale' (QMS), a value m(z₀; A) qq X may be treated as a measure for the quality 'membership in the fuzzy set A'. For the generalized membership function an universal form is found, namely the form of the universal membership function u(z; A): Z → X′, which maps the fixed set Z into mathematical system S′ = (X′; R_qq), the mathematical system S′ being an universal representation for the initial system S = (X; R). The structure R_qq of the universal mathematical system is formed from so-called chain-dominance r(i)-adic relations R_qq ^r(i) (R_≥ ⁽ⁱ⁾), which are induced by corresponding order relations R_≥ ⁽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.