Subgradients of algebraically convex functions: a Galois connection relating convex sets and subgradients of convex functions
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Subgradients of convex functions have been defined in classical analysis for the Euclidean spaces. Here we describe a Galois connection between convex upsets and maximal homomorphisms (which are subgradient values), and interpret traditional results in terms of universal algebra, noting that (IR[superscript]n, I°) is a mode and (IR, max, I°) is a modal. These traditional results are then generalized to give subgradients of convex functions from modes to modals, and necessary conditions for existence are explored.