Insertion and extension theorems for lattice-valued functions on preordered topological spaces

In this paper, the possibility of inserting an increasing continuous lattice-valued function between two comparable semicontinuous functions is studied. First, we obtain a sufficient condition for such insertion, then we get new characterizations of several classes of preordered topological spaces, among them normally preordered and extremally preorder-disconnected spaces. Conditions for the continuous and increasing extension of lattice-valued maps of the same type defined on closed (resp. open) sets are also investigated.