Order Isomorphism

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An order isomorphism is a particular type of isomorphism that preserves order.

We say that $f$ creates an isomorphism between two relational systems $( A , <_A )$ and $( B , <_B )$ if and only if $f$ creates a bijection between $A$ and $B$ and for all $x$ and $y$ in $A$, $x <_A y \leftrightarrow f(x) <_B f(y)$

Properties

Order-isomorphisms preserve ordering, so if $( A , <_A )$ is strictly ordered, founded, or well-ordered, then $( B , <_B )$ will be as well.

All well-ordered sets are isomorphic to a unique ordinal. If two ordinals are order-isomorphic with respect to membership, then they are equal. Between two well-ordered sets $A$ and $B$, exactly 1 of the following will hold:

  • $A$ is order-isomorphic to $B$
  • $A$ is order-isomorphic to an initial segment of $B$
  • An initial segment of $A$ is order-isomorphic to $B$