Order isomorphism

In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.

Formally, given two posets S=(S, ≤_S) and T=(T, ≤_T) an order isomorphism from (S, ≤_S) to (T, ≤_T) is a surjective function h : S → T such that for all u and v in S,

h(u) ≤_T h(v) if and only if u ≤_S v.

In this case, the posets S and T are said to be order isomorphic. Note that the above definition characterizes order isomorphisms as surjective order embeddings. It should also be remarked that order isomorphisms are necessarily injective. Hence, yet another characterization of order isomorphisms is possible: they are exactly those monotone bijections that have a monotone inverse.

An order isomorphism from (S, ≤) to itself is called an order automorphism.

Examples

• Negation is an order isomorphism from (R,≤) to (R,≥), since -x ≥ -y if and only if x ≤ y.
• The function f(x) = x-1 is an order automorphism on (R,≤), since x-1 ≤ y-1 if and only if x ≤ y.
• A utility function is an order isomorphism from some consumption set into the real line.

See also

• Order type