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 ${\displaystyle (S,\leq _{S})}$ and ${\displaystyle (T,\leq _{T})}$, an order isomorphism from ${\displaystyle (S,\leq _{S})}$ to ${\displaystyle (T,\leq _{T})}$ is a bijective function ${\displaystyle f}$ from ${\displaystyle S}$ to ${\displaystyle T}$ with the property that, for every ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle S}$, ${\displaystyle x\leq _{S}y}$ if and only if ${\displaystyle f(x)\leq _{T}f(y)}$. That is, it is a bijective order-embedding.

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