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In the category of partially ordered sets, the morphisms usually considered are (weakly) increasing maps, that is if $(P_1,\leq_1)$ and $(P_2,\leq_2)$ are partial orders, then a map $f:P_1\to P_2$ is a morphism if $x\leq_1 y$ implies $f(x)\leq_2 f(y)$, for all $x,y\in P_1$.

In model theory and logic, we often consider strong homomorphisms (also called reductions) of relation structures, which in this case would be maps $f:P_1\to P_2$ such that $x\leq_1 y$ if and only if $f(x)\leq_2 f(y)$, for all $x,y\in P_1$. In other words, such maps not only preserve order, but preserve non-order as well.

A simple question: Do such maps have a standard name in the literature on partially ordered sets?

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the term is exactly : "order-embedding". Such functions are necessarily injective, btw! More generally, embeddings in model theory both preserve and reflect relations.

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    $\begingroup$ Yes, preserve a reflect order are the usual terms, not "preserving order and non-order". $\endgroup$
    – amrsa
    Jan 11 at 18:34

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