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 either 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 and , an order isomorphism from to is a bijective function from to with the property that, for every and in , if and only if . That is, it is a bijective order-embedding. It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that cover all the elements of and that it preserve orderings, are enough to ensure that is also one-to-one, for if then (by the assumption that preserves the order) it would follow that and , implying by the definition of a partial order that . Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse. An order isomorphism from a partially ordered set to itself is called an order automorphism. When an additional algebraic structure is imposed on the posets and , a function from to must satisfy additional properties to be regarded as an isomorphism. For example, given two partially ordered groups (po-groups) and , an isomorphism of po-groups from to is an order isomorphism that is also a group isomorphism, not merely a bijection that is an order embedding. The identity function on any partially ordered set is always an order automorphism. Negation is an order isomorphism from to (where is the set of real numbers and denotes the usual numerical comparison), since −x ≥ −y if and only if x ≤ y. The open interval (again, ordered numerically) does not have an order isomorphism to or from the closed interval : the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements.