# Is there a name for order-preserving functions $f$ where “$a\le b$ if and only if $f(a) \le f(b)$”? [closed]

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 $$\mathbb{N}$$).

• The first sentence is really an understatement: in fact, this is much stronger than monotonicity. – Jochen Glueck Jun 30 at 5:30
• 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. – YCor Jun 30 at 14:36

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.