Orderreversing

Order-reversing, or antitone, describes a function between partially ordered sets that inverts the order. Formally, a function f: (P, ≤P) → (Q, ≤Q) is order-reversing if for all x, y in P, x ≤P y implies f(x) ≥Q f(y). Equivalently, f is order-preserving when viewed as a map from P to the dual poset Q^op, where the order relation is reversed.

Key properties include how order-reversing maps compose. The composition of two order-reversing maps is order-preserving. The composition of an order-preserving map with an order-reversing map is order-reversing, and vice versa. This perspective highlights the compatibility of order-reversing maps with duality: they become monotone when either the domain or codomain is considered with reversed order.

Examples help anchor the concept. On the real numbers with the usual ≤, the negation map f(x) = -x is order-reversing, since x ≤ y implies -x ≥ -y. On the positive reals (0, ∞), the reciprocal map f(x) = 1/x is order-reversing as x < y implies 1/x > 1/y. In a finite boolean lattice, the logical complement x ↦ ¬x is order-reversing. In lattice theory, order-reversing maps are called antitone maps and frequently appear in dualities and Galois connections, where one side is order-preserving and the other order-reversing, and in constructing monotone operators by composing with themselves.