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Homestructure homomorphism
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structure homomorphism
Let $\Sigma$ be a fixed signature, and $\mathfrak{A}$ and $\mathfrak{B}$ be two structures for $\Sigma$. The interesting functions from $\mathfrak{A}$ to $\mathfrak{B}$ are the ones that preserve the structure.
A function $f\colon\mathfrak{A}\to\mathfrak{B}$ is said to be a homomorphism (or simply morphism) if and only if:
1. For every constant symbol $c$ of $\Sigma$, $f(c^{\mathfrak{A}})=c^{\mathfrak{B}}$.
2. For every natural number $n$ and every $n$ary function symbol $F$ of $\Sigma$,
$f(F^{\mathfrak{A}}(a_{1},...,a_{n}))=F^{\mathfrak{B}}(f(a_{1}),...,f(a_{n})).$ 3. For every natural number $n$ and every $n$ary relation symbol $R$ of $\Sigma$,
$R^{\mathfrak{A}}(a_{1},\ldots,a_{n})\Rightarrow R^{\mathfrak{B}}(f(a_{1}),% \ldots,f(a_{n})).$
Homomorphisms with various additional properties have special names:

An injective homomorphism is called a monomorphism.

A surjective homomorphism is called an epimorphism.

A bijective homomorphism is called a bimorphism.

A surjective embedding is called an isomorphism.

A homomorphism from a structure to itself (e.g., $f\colon\mathfrak{A}\to\mathfrak{A}$) is called an endomorphism.

An isomorphism from a structure to itself is called an automorphism.
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