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This is something only slightly stronger than monotonicity. I think that in category theory this would be a fully faithful functor, but I’m not sure if there is a standard name for this in order theory.

For context, I’m working in a theorem prover (Isabelle) and this is a property of the function which converts words to natural numbers (embedding $\{0 \ldots 2^{32} - 1\}$ into $\mathbb{N}$).

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  • $\begingroup$ The first sentence is really an understatement: in fact, this is much stronger than monotonicity. $\endgroup$ – Jochen Glueck Jun 30 at 5:30
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    $\begingroup$ For posets this condition forces injectivity so "order embedding" is natural. But for preorders (=set with reflexive transitive relations), it doesn't imply injective, so "order embedding" would be misleading. $\endgroup$ – YCor Jun 30 at 14:36
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The first thing I thought of was

order embedding

and this is confirmed by an article on monotonicity in order theory.

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The first thing I thought of was also order embedding (like Bjørn), but one can also say that $f$ preserves and reflects the order. That way, you understand even if you don't know what an order embedding is. Folks sometimes write that $f$ is a strong order homomorphism in this situation.

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    $\begingroup$ I rather think that “order embedding” is more clear even if you see the terms for the first time; most people do not know what “reflects” means, whereas embeddings are commonplace. However, the “preserves and reflects” or “strong homomorphism” terminology has one big advantage, namely that it is applicable more generally to any relational structures, whereas the identification of this property with order embedding relies on reflexivity and antisymmetry of the partial order, and it is not valid in general. $\endgroup$ – Emil Jeřábek Jun 30 at 6:01
  • $\begingroup$ @EmilJeřábek Somehow, your comment makes me think of languages without equality. $\endgroup$ – Bjørn Kjos-Hanssen Jun 30 at 6:12

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