202601171959

Tags : Model Theory

Homomorphism of Models

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On fixing a language $L$, given 2 $L$-models $\mathcal{M},\mathcal{N}$, we define a model homomorphism $\hat{h}:\mathcal{M}\to \mathcal{N}$ as:

Definition

The homomorphism $\hat{h}$ carries the following data:

• $h:M\to N$
• For any constant symbol $c$ we have $h(c^{\mathcal{M}})=c^{\mathcal{N}}$
• For any function symbol $f^{(n)}$ we have $h(f^{\mathcal{M}}(\bar{a}))=f^{\mathcal{N}}(h(\bar{a}))$
• For any relation symbol $r^{(n)}$ we have $r^{\mathcal{M}}(\bar{a})\Rightarrow r^{\mathcal{N}}(h(\bar{a}))$

This does define a Category.

Such a morphism is called Strict if for any relation symbol $r^{(n)}$, and for any witness $\bar{a}\in h(M)$ such that $r^{\mathcal{N}}(\bar{a})$, there is a witness $\bar{b}\in M$ such that $\bar{b}\in r^{\mathcal{M}}$.

This also defines a category.

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References

• Semantics of First Order Logic