A New Extension Theorem for Concave Operators

We present a new and interesting extension theorem for concave operators as follows. Let X be a real linear space, and let Y,K be a real order complete PL space. Let the setA ⊂ X × Y be convex. Let X0 be a real linear proper subspace of X, with θ ∈ AX −X0 , where AX {x | x, y ∈ A for some y ∈ Y}. Let g0 : X0 → Y be a concave operator such that g0 x ≤ z whenever x, z ∈ A and x ∈ X0. Then there exists a concave operator g : X → Y such that i g is an extension of g0, that is, g x g0 x for all x ∈ X0, and ii g x ≤ z whenever x, z ∈ A.