On hyperplanes and semispaces in max-min convex geometry

Consider the set B = [0, 1] endowed with the operations⊕ = max,∧ = min. This is a well-known distributive lattice, and like any distributive lattice it can be considered as a semiring equipped with addition ⊕ and multiplication ⊗ := ∧. Importantly, both operations are idempotent, a⊕a = a and a⊗a = a∧a = a, and closely related to the order: a⊕ b = b ⇔ a ≤ b ⇔ a∧ b = a. For standard literature on lattices and semirings see e. g. [1] and [9]. We consider B, the cartesian product of n copies of B, and equip this cartesian product with operations of taking componentwise ⊕: (x⊕y)i := xi⊕yi for x, y ∈ B n and i = 1, . . . , n, and scalar ∧-multiplication: (a ∧ x)i := a ∧ xi for a ∈ B, x ∈ B n and i = 1, . . . , n. Thus B is considered as a semimodule over B [9]. Alternatively, one may think in terms of vector lattices [1]. A subset C of B is said to be max-min convex, (or briefly convex), if the relations x, y ∈ C,α, β ∈ B, α⊕ β = 1 imply α ∧ x ⊕ β ∧ y ∈ C. Here and everywhere in the paper we assume the priority of ∧ over ⊕. If x, y ∈ B, the set [x, y] := {α ∧ x⊕ β ∧ y ∈ B|α, β ∈ B, α⊕ β = 1} = {max(min(α, x),min(β, y)) ∈ B|α, β ∈ B,max (α, β) = 1}, (1)