On lattice-valued frames: The completely distributive case

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a completely distributive lattice. In particular, L-Frm is complete and cocomplete. Finally we prove that, in a certain sense, the assumption of L being a completely distributive lattice cannot be weakened.