# proof of homomorphic image of a $\Sigma$-structure is a $\Sigma$-structure

We need to show that $\operatorname{im}(f)$ is closed under functions. For every constant symbol $c$ of $\Sigma$, $c^{\mathfrak{B}}=f(c^{\mathfrak{A}})$. Hence $c^{\mathfrak{B}}\in\operatorname{im}(f)$. Also, if $b_{1},\ldots,b_{n}\in\operatorname{im}(f)$ and $F$ is an $n$-ary function symbol of $\Sigma$, then for some $a_{1},\ldots,a_{n}\in\mathfrak{A}$ we have

 $F^{\mathfrak{B}}(b_{1},\ldots,b_{n})=F^{\mathfrak{B}}(f(a_{1}),\ldots,f(a_{n})% )=f(F^{\mathfrak{A}}(a_{1},\ldots,a_{n})).$

Hence $F^{\mathfrak{B}}(b_{1},\ldots,b_{n})\in\operatorname{im}(f)$.

Title proof of homomorphic image of a $\Sigma$-structure is a $\Sigma$-structure ProofOfHomomorphicImageOfASigmastructureIsASigmastructure 2013-03-22 13:46:47 2013-03-22 13:46:47 almann (2526) almann (2526) 4 almann (2526) Proof msc 03C07