User:Zero sharp/Maps between structures

Maps between structures
Fix a language, $$L$$ and let $$M$$ and $$N$$ be two $$L$$-structures. For symbols from the language, such as a constant $$c$$, let $$c^M$$ be the interpretation of $$c$$ in $$M$$ and similarly for the other classes of symbols (functions and relations).

A map $$j$$ from the domain of $$M$$ to the domain of $$N$$ is a homomorphism if the following conditions hold:


 * 1) for every constant symbol $$c \in L$$,  we have $$j(c^M) = c^N$$.
 * 2) for every n-ary function symbol $$f \in L$$ and $$a_1,\ldots,a_n \in M^n$$, we have $$j(f^M(a_1,\ldots,a_n))=f^N(j(a_1),\ldots,j(a_n))$$,
 * 3) for every n-ary relation symbol $$R \in L$$ and $$a_1,\ldots,a_n \in M^n,$$ we have $$M \models R(a_1,\ldots,a_n) \Rightarrow N \models R(j(a_1),\ldots,j(a_n))$$,

If in addition, the map $$j$$ is injective and the third condition is modified to read:


 * for every n-ary relation symbol $$R \in L$$ and $$a_1,\ldots,a_n \in M^n,$$ we have $$M \models R(a_1,\ldots,a_n) \Leftrightarrow N \models R(j(a_1),\ldots,j(a_n)),$$

then the map $$j$$ is an embedding (of $$M$$ into $$N$$).

Equivalent definitions of homomorphism and embedding are:

If for all atomic formulas $$\phi$$ and sequences of elements from $$M$$, $$\bar{a} = (a_1,a_2,\ldots,a_n)$$
 * $$M \models \phi [\bar{a}] \Rightarrow N  \models  \phi [\bar{b}]$$

where $$\bar{b}$$ is the image of $$\bar{a}$$ under $$j$$:
 * $$\bar{b} = (b_1,b_2,\ldots,b_n) = (j(a_1),j(a_2),\ldots,j(a_n)) = j(\bar{a})$$

then $$j$$ is a homomorphism. If instead:


 * $$M \models \phi [\bar{a}] \Leftrightarrow N  \models  \phi [\bar{b}]$$

then $$j$$ is an embedding.