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

We need to show that $\mathrm{im}(f)$ is closed under^{} functions. For every constant symbol $c$ of $\mathrm{\Sigma}$, ${c}^{\U0001d505}=f({c}^{\U0001d504})$. Hence ${c}^{\U0001d505}\in \mathrm{im}(f)$. Also, if ${b}_{1},\mathrm{\dots},{b}_{n}\in \mathrm{im}(f)$ and $F$ is an $n$-ary function symbol of $\mathrm{\Sigma}$, then for some ${a}_{1},\mathrm{\dots},{a}_{n}\in \U0001d504$ we have

$${F}^{\U0001d505}({b}_{1},\mathrm{\dots},{b}_{n})={F}^{\U0001d505}(f({a}_{1}),\mathrm{\dots},f({a}_{n}))=f({F}^{\U0001d504}({a}_{1},\mathrm{\dots},{a}_{n})).$$ |

Hence ${F}^{\U0001d505}({b}_{1},\mathrm{\dots},{b}_{n})\in \mathrm{im}(f)$.

Title | proof of homomorphic image^{} of a $\mathrm{\Sigma}$-structure^{} is a $\mathrm{\Sigma}$-structure |
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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 |