Cuts of poset-valued functions in the framework of residuated maps

Abstract

We analyze cuts of poset-valued functions relating them to residuated mappings. Dealing with the lattice-valued case we prove that a function μ:X→L induces a residuated map f:L→P(X) whose values are the cuts of μ and we describe the corresponding residual. Conversely, it turns out that every residuated map f from L to the power set of X determines a lattice valued function μ:X→L whose cuts coincide with the values of f. For general poset-valued functions, we give conditions under which the map sending an element of a poset to the corresponding cut is quasi-residuated, and then conditions under which it is also residuated. We prove that without additional conditions, the map analogue to the residual is a partial function hence we get particular weakly residuated maps which, on the power set of the domain, generate centralized systems instead of closures. We show that the main properties of residuated maps are preserved in this generalized case. We apply these results to the canonical representation of poset-valued and lattice-valued functions, using the corresponding closures and centralized systems.