Poset valued convexities

We analyze cut properties of lattice and poset valued functions (fuzzy sets) of n-dimensional real variable with respect to convexity. In the lattice valued case, convexity of a fuzzy set is equivalent with the convexity of its standard cuts. Dealing with non standard cuts, we show that unless the space of the values is a chain, such statement in general does not hold. We present analogue results in the more general setting of poset valued functions. We also prove that there exist lattice and poset valued functions whose cuts are precisely the convex sets of particular families. Our work is motivated by possible applications of convex functions (e.g., in image processing).