Elementary topical functions on \(b\)-complete semimodules over \(b\)-complete idempotent semifields (Q603143)

Let \(X\) be a \(b\)-complete semimodule over a \(b\)-complete idempotent semifield \(K\), both equipped with the order \(x\leq y\iff x\oplus y=y\). A function \(f: X\to K\) is called topical if it is homogeneous and increasing. The paper characterizes topical functions as the pointwise maxima of so-called elementary topical functions. In the case \(X=R^n\), \(x\oplus y=\max(x,y)\), the elementary topical functions are the so-called ``min-type'' functions; they are homogeneous and commute with the \(\min\)-operation. As an application, under some compatibility hypotheses on the topologies, it is shown that downward sets in \(X\) are closed if and only if they are closed along rays, and that these sets are characterized by a ``separation'' property of elementary topical functions.