Subdifferentiation of monotone functions from semilattices to distributive lattices (Q1904389)

A monotone function \(f\) from a semilattice \(S\) to a distributive lattice \(D\) is an example of a convex function from a mode to a modal [see \textit{A. B. Romanowska} and \textit{J. D. H. Smith}, Modal theory: an algebraic approach to order, geometry and convexity (1985; Zbl 0553.08001), and \textit{A. Romanowska}, ``An introduction to the theory of modes and modals'', Contemp. Math. 131, Part 3, 241-262 (1992; Zbl 0776.08003) for the general theory of modals]. The main theorem of the paper says that \(f\) is a join of semilattice homomorphisms \(S \to D\), provided that \(D\) is complete. The following definition, which generalizes algebraically the notion of a subgradient of a convex real function, reveals the connections of these homomorphisms with subdifferentiation: for any \(c \in S\), the subgradient of \(f\) at \(c\) is the set \(\{k \in \Hom ((S, \wedge), (D, \wedge)) \mid k\leq f,k (c) = f(c)\}\).