The Relationships Between KM-fuzzy Quasi-metric Spaces and the Associated Posets of Formal Balls

The purpose of this paper is to investigate the connections between KM-fuzzy quasi-metric spaces and the associated order structures of formal balls. We introduce the notions of Yoneda T-completeness and Yoneda S-completeness on KM-fuzzy quasi-metric spaces. In both cases, the associated posets of formal balls are directed complete. In particular, for a subclass of KM-fuzzy quasi-metric spaces, Yoneda T-completeness and Yoneda S-completeness are characterized respectively by directed completeness of the associated posets of formal balls.