The Meet-continuity of L-semilattices

This paper is devoted to search for Cartesian closed category of quantitative domain categories. Firstly, inspired by the method of constructing L-frames, the meet-continuity on L-semilattices is built and characterized, which generalizes the meet-continuity on crisp semilattices. Then, it is shown that a complete L-lattice is an L-frame iff it is distributive and meet-continuous. In particular, it is shown that the category L-MC of meet-continuous L-lattices is cartesian closed.