Formal balls in fuzzy quasi-metric spaces

The notions of Yoneda completeness and Smyth completeness on fuzzy quasi-metric spaces are introduced and their relationship with other types of completeness including sequentially Yoneda completeness and bicompleteness are investigated. Then we use the standard Yoneda completeness to characterize the order-theoretical properties of the poset (BX, \sqsubseteqM) of formal balls in a fuzzy quasi-metric space (X,M, ∧). The results show that if (BX, \sqsubseteq M) is a dcpo, then (X,M, ∧) is standard complete and conversely, (BX, \sqsubseteqM) forms a dcpo provided that (X,M, ∧) is standard Yoneda complete. Particularly, in a fuzzy metric space, we clarify three types of completeness which can be characterized by the directed completeness of the related poset of formal balls.