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
Canonical name	ProofOfHomomorphicImageOfASigmastructureIsASigmastructure
Date of creation	2013-03-22 13:46:47
Last modified on	2013-03-22 13:46:47
Owner	almann (2526)
Last modified by	almann (2526)
Numerical id	4
Author	almann (2526)
Entry type	Proof
Classification	msc 03C07

proof of homomorphic image of a Σ-structure is a Σ-structure

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