Meet-continuity on $L$-directed Complete Posets

In this paper, the definition of meet-continuity on $L$-directed complete posets (for short, $L$-dcpos) is introduced. As a generalization of meet-continuity on crisp dcpos, meet-continuity on $L$-dcpos, based on the generalized Scott topology, is characterized. In particular, it is shown that every continuous $L$-dcpo is meet-continuous and $L$-continuous retracts of meet-continuous $L$-dcpos are also meet-continuous. Then, some topological properties of meet-continuity on $L$-dcpos are discussed. It is shown that meet-continuity on $L$-dcpos is a topological invariant with respect to the generalized Scott topology, and meet-continuity on $L$-dcpos is hereditary with respect to generalized Scott closed subsets.