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Encyclopedia > 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. Whenever two partially ordered sets 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 partially ordered sets (S, ≤) and (T, <=) an order isomorphism from (S, ≤) to (T, <=) is a surjective function h : ST such that for all u and v in S,

h(u) <= h(v) iff uv.

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. Results from FactBites:

 Santa Fe Institute Research Topics - Information Processing & Computation in Complex Systems (1927 words) The other is the many-atomic order of statistical mechanics, where ordered states are derived through an appropriate averaging over many interacting particles. In order to build cells capable of responding adaptively to environmental cues, elaborate signaling networks have evolved capable of reliably, transducing energy from the environment into internal informational states of the cell. Robustness arises as a byproduct of statistical order in biology as individual components are frequently correlated in their activity, and this can be exploited as a source of redundancy.
 Order-embedding - definition of Order-embedding in Encyclopedia (665 words) In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Yet, the two posets are not isomorphic: [−1,1] has both a least and a greatest element, which are not present in the case of the real numbers. The basic category for the study of partial orders is the category of posets and monotone functions.
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