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.^[1]

The idea of isomorphism can be understood for finite orders in terms of Hasse diagrams. Two finite orders are isomorphic exactly when a single Hasse diagram (up to relabeling of its elements) expresses them both, in other words when every Hasse diagram of either can be converted to a Hasse diagram of the other by simply relabeling the vertices.