Partial ordering in L-underdeterminate sets

The purpose of this paper is to improve results on fuzzy partial orderings obtained by Zadeh in [9]. We overcome the difficulties connected with the axioms of antisymmetry and linearity. Moreover, if the underlying lattice L is a complete residuated lattice, we establish a Szpilrajn theorem, i.e., any (L-fuzzy) partial ordering has a linear extension. In opposition to Zadeh's, our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless. Therefore we first introduce the category LUS (cf. [2]), which can be considered as a mathematical model of fuzzy equality, and subsequently we specify the axioms of (L-fuzzy) partial orderings with respect to the frame given by LUS. The axioms we use clearly display the usefulness of having a Zadeh-like complementation and, as a consequence, the usefulness of a positivistic (and nonintuitionistic) frame of study. An example concerning L°(Rn) which we give clearly shows that the LUS version of the Szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem, but provides us with additional information.