Yoneda completeness and flat completeness of ordered fuzzy sets
arXiv:1607.03212 · doi:10.1016/j.fss.2016.06.009
Abstract
This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed completeness in order theory to the fuzzy setting. Yoneda completeness requires every forward Cauchy net converges (has a Yoneda limit), while flat completeness requires every flat weight (a counterpart of ideals in partially ordered sets) has a supremum. It is proved that flat completeness implies Yoneda completeness, but, the converse implication holds only in the case that the related triangular norm is either isomorphic to the Lukasiewicz t-norm or to the product t-norm.
25 pages in Fuzzy Sets and Systems (2016)