Meet-continuity on $L$-directed Complete Posets

In this paper, the definition of meet-continuity on $L$-directedcomplete posets (for short, $L$-dcpos) is introduced. As ageneralization of meet-continuity on crisp dcpos, meet-continuity on$L$-dcpos, based on the generalized Scott topology, ischaracterized. In particular, it is shown that every continuous$L$-dcpo is meet-continuous and $L$-continuous retracts ofmeet-continuous $L$-dcpos are also meet-continuous. Then, sometopological properties of meet-continuity on $L$-dcpos arediscussed. It is shown that meet-continuity on $L$-dcpos is atopological invariant with respect to the generalized Scotttopology, and meet-continuity on $L$-dcpos is hereditary withrespect to generalized Scott closed subsets.