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# Order isomorphism  ( Order Theory )

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In the field of an order isomorphism is a special kind of monotone function that constitutes a suitable notion of 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 and Galois connections.; .

Definition
Formally, given two posets $\left(S,\le_S\right)$ and $\left(T,\le_T\right)$, an order isomorphism from $\left(S,\le_S\right)$ to $\left(T,\le_T\right)$ is a $f$ from $S$ to $T$ with the property that, for every $x$ and $y$ in $S$, $x \le_S y$ if and only if $f\left(x\right)\le_T f\left(y\right)$. That is, it is a bijective .This is the definition used by . For and it is a consequence of a different definition.

It is also possible to define an order isomorphism to be a order-embedding. The two assumptions that $f$ cover all the elements of $T$ and that it preserve orderings, are enough to ensure that $f$ is also one-to-one, for if $f\left(x\right)=f\left(y\right)$ then (by the assumption that $f$ preserves the order) it would follow that $x\le y$ and $y\le x$, implying by the definition of a partial order that $x=y$.

Yet another characterization of order isomorphisms is that they are exactly the monotone that have a monotone inverse.This is the definition used by and .

An order isomorphism from a partially ordered set to itself is called an order ., p. 13.

When an additional algebraic structure is imposed on the posets $\left(S,\le_S\right)$ and $\left(T,\le_T\right)$, a function from $\left(S,\le_S\right)$ to $\left(T,\le_T\right)$ must satisfy additional properties to be regarded as an isomorphism. For example, given two partially ordered groups (po-groups) $\left(G, \le_G\right)$ and $\left(H, \le_H\right)$, an isomorphism of po-groups from $\left(G,\leq_G\right)$ to $\left(H,\le_H\right)$ is an order isomorphism that is also a group isomorphism, not merely a bijection that is an .This definition is equivalent to the definition set forth in .

Examples
• The identity function on any partially ordered set is always an order automorphism.
• is an order isomorphism from $\left(\mathbb\left\{R\right\},\leq\right)$ to $\left(\mathbb\left\{R\right\},\geq\right)$ (where $\mathbb\left\{R\right\}$ is the set of and $\le$ denotes the usual numerical comparison), since − x ≥ − y if and only if xy.See example 4 of , p. 39., for a similar example with integers in place of real numbers.
• The $\left(0,1\right)$ (again, ordered numerically) does not have an order isomorphism to or from the $0,1$: the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements., example 1, p. 39.

Order types
If $f$ is an order isomorphism, then so is its . Also, if $f$ is an order isomorphism from $\left(S,\le_S\right)$ to $\left(T,\le_T\right)$ and $g$ is an order isomorphism from $\left(T,\le_T\right)$ to $\left(U,\le_U\right)$, then the function composition of $f$ and $g$ is itself an order isomorphism, from $\left(S,\le_S\right)$ to $\left(U,\le_U\right)$.; .

Two partially ordered sets are said to be order isomorphic when there exists an order isomorphism from one to the other.. Identity functions, function inverses, and compositions of functions correspond, respectively, to the three defining characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into equivalence classes, families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called .

• Permutation pattern, a permutation that is order-isomorphic to a subsequence of another permutation

Notes
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