What is a practicized (or suitable) label for morphisms that satisfy the condition mentioned in the body of this question?

Question

Let it be that $\mathfrak A$ and $\mathfrak B$ denote $\mathcal L$-models for some language $\mathcal L$ and that $A$ and $B$ denote their domains/universes.

A function $h:A\to B$ can be recognized as a morphism $\mathfrak A\to\mathfrak B$ if it satisfies the conditions:

• $h(c^{\mathfrak A})=c^{\mathfrak B}$ for constants symbols.
• $h\circ f^{\mathfrak A}=f^{\mathfrak B}\circ h^n$ for function symbols $f$.
• $(a_1,\dots,a_n)\in r^{\mathfrak A}\implies (h(a_1),\dots,h(a_n))\in r^{\mathfrak B}$ for relation symbols $r$.

In both cases $n$ denotes the arity of the symbol.

If moreover $h$ is injective and satisfies the stronger condition:

$$(a_1,\dots,a_n)\in r^{\mathfrak A}\iff (h(a_1),\dots,h(a_n))\in r^{\mathfrak B}\text{ for relation symbols }r\tag1$$then $h$ is an embedding.

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My question:

Is there some suitable terminology for a morphism $h$ that satisfies $(1)$?

I would like to say things as: "a morphism $h$ is an embedding if it is injective and ...."

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Edit:

The answer of Alex is fine and educating. I have found a more direct answer to my question later in some script. Homomorphisms that satisfy $(1)$ were called strong homomorphisms there. So a homomorphism is an embedding iff it is strong and injective. Also I was told there that strong homomorphisms correspond one-to-one with congruences (hence quotients).

Answer 1

The following terminology is commonly used. For an $L$-formula $\varphi(x_1,\dots,x_n)$, $L$-structures $A$ and $B$, and a function $h\colon A\to B$:

• $h$ preserves $\varphi$ means that for all $a_1,\dots,a_n\in A$, if $A\models \varphi(a_1,\dots,a_n)$, then $B\models \varphi(h(a_1),\dots,h(a_n))$.
• $h$ reflects $\varphi$ means that for all $a_1,\dots,a_n\in A$, if $B\models \varphi(h(a_1),\dots,h(a_n))$, then $A\models \varphi(a_1,\dots,a_n)$.

This terminology can also be applied to relation symbols, where we identify an $n$-ary relation symbol $R$ with the atomic formula $R(x_1,\dots,x_n)$.

You can also apply it to function symbols (and constant symbols, which are are just $0$-ary function symbols), by identifying an $n$-ary function symbol $f$ with the atomic formula $f(x_1,\dots,x_n) = y$. But since function symbols have a different character than relation symbols, I usually prefer to use the equivalent definition suggested in your question: $h$ preserves (or respects) $f$ if $h\circ f^A = f^B\circ h^n$.

Using this terminology, a homomorphism is a function which preserves the symbols in the language. An embedding is a homomorphism which is injective and reflects the relation symbols in the language.

A more elegant characterization is that a homomorphism is a function which preserves all atomic formulas, and an embedding is a function which preserves and reflects all atomic formulas.