Homomorphism

• Let $\mathcal{M}$ and $\mathcal{N}$ model in $L$, and $H:\mathcal{M}\to \mathcal{N}$ a function

  • We say that $H$ preserves the relations if for every $n$-ary predicate symbol $R$ in $L$, and for all $a_1,\dots ,a_n$ elements of the domain, we have:
    • $(a_1,\dots ,a_n)\in R^{\mathcal{M}}⟹(H(a_1),\dots ,H(a_n))\in R^{\mathcal{N}}$
  • We say that $H$ reflects the relations if for every $n$-ary predicate symbol $R$ in $L$, and for all $a_1,\dots ,a_n$ elements of the domain, we have:
    • $(a_1,\dots ,a_n)\in R^{\mathcal{M}}⟸(H(a_1),\dots ,H(a_n))\in R^{\mathcal{N}}$
  • $H$ is a homomorphism of $\mathcal{M}$ into $\mathcal{N}$ if:
    • ($H$ preserves constants) For every term $c$ in $L$, we have $H(c^{\mathcal{M}})=c^{\mathcal{N}}$
      • (todo in other book this goes on constants only)
    • ($H$ preserves functions) For every $n$-ary function symbol $f$ and for all $a_1,\dots ,a_n$ elements of the domain $A$ we have:
      • $H(f^{\mathcal{M}}(a_1,\dots ,a_n))=f^{\mathcal{N}}(H(a_1),\dots ,H(a_n))$
    • $H$ preserves relations
  • $H$ is a strong homomorphism if:
    • $H$ is a homomorphism
    • $H$ reflects relations
  • $H$ is an embedding if:
    • $H$ is a homomorphism
    • $H$ is injective (one-to-one)
    • $H$ reflects relations
  • $H$ is an isomorphism if:
    • $H$ is a homomorphism
    • $H$ is injective (one-to-one)
    • $H$ is surjective (onto) $\mathcal{N}$
    • $H$ reflects relations
  • $\mathcal{M}$ and $\mathcal{N}$ are isomorphic if an isomorphism exists between them
  • $H$ is called an epimorphism if:
    • $H$ is a homomorphism
    • $H$ is surjective (onto) $\mathcal{N}$
    • $H$ reflects relations (possibly excluding the equality relation)
  • $H$ is called an automorphism if:
    • $H$ is an isomorphism
    • $\mathcal{M}=\mathcal{N}$
  • $H$ is called an endomorphism if:
    • $H$ is a homomorphism
    • $\mathcal{M}=\mathcal{N}$
  • $H$ is called a monomorphism if:
    • $H$ is a homomorphism
    • $H$ is injective (one-to-one)

• The model $\mathcal{N}$ is a (strong) homomorphic image of the model $\mathcal{M}$ if there exists a (strong) homomorphism from $\mathcal{M}$ onto $\mathcal{N}$

• If $H$ is an epimorphism that reflects also the equality relation, then $H$ is calledtodo

In algebra, when dealing with groups, rings, and linear spaces, the focus is on functions (operations) and not on relations, therefore the condition $H$ reflects relations is not relevant, therefore, for example, a surjective homomorphism is an epimorphism

If one of the relation in $L$ is the equality relation, (and it is the real equality in $\mathcal{M}$ and $\mathcal{N}$), then by the property ”$H$ reflects relations” we have that $H$ is injective. Therefore, in this case, an epimorphism is an isomorphism.

• (6.7) (assume $L$ with equality) Let $H:\mathcal{M}\to \mathcal{N}$ be an isomorphism, and $\phi [x_1,\dots ,x_n]$ a formula (i.e. a formula with free variables $x_1,\dots ,x_n$), and let $a_1,\dots ,a_n$ elements of the domain of $\mathcal{M}$. Then $\mathcal{M}\models \phi [a_1,\dots ,a_n]⟺\mathcal{N}\models \phi [H(a_1),\dots ,H(a_n)]$

draft

https://people.math.sc.edu/mcnulty/762/modeltheory.pdf

Congruence Relation

• A binary relation $E$ on a model $M$ in $L$ is called a congruence (relation) on $M$ if

  • $E$ is an equivalence relation
  • if $f$ is n-ary function in $L$, and $(b_1,a_1)\in E,\dots ,(b_n,a_n)\in E$, then $(f^M(b_1,\dots ,b_n),f^M(a_1,\dots ,a_n))\in E$
  • if $R$ is n-ary relation in $L$, and $(b_1,a_1)\in E,\dots ,(b_n,a_n)\in E$, then $(b_1,\dots ,b_n)\in R⟺(a_1,\dots ,a_n)\in R$

• Let be a model $M=⟨D,\dots ⟩$ in $L$.

  • We define a model $M^{\prime }=⟨D^{\prime },\dots ⟩$ as follow:
    • $D^{\prime }$ is the quotient set of $D$ induced by $E$
    • The function $I:D\to D^{\prime }$, $I(a)=[a]$ is called (ההתאמה הטבעית)
    • For every constant $c$ in $L$, we define $c^{M^{\prime }}=[c^M]$
    • Let be n-ary function symbol $f$ in $L$, and $C_1,\dots ,C_n$ the equivalence classes in $M^{\prime }$, we choose $a_1,\dots ,a_n$ where $a_i\in C_i$, and we define $f^{M^{\prime }}(C_1,\dots ,C_n)$ to be the equivalence class in which $f^M(a_1,\dots ,a_n)$ is contained. (i.e. $f^{M^{\prime }}([a_1],\dots ,[a_n])=[f^M(a_1,\dots ,a_n)]$)
    • For every n-ary relation $R$ in $L$, we define the relation $R^{M^{\prime }}$ in $M^{\prime }$ as follow: $([a_1],\dots ,[a_n])\in R^{M^{\prime }}⟺(a_1,\dots ,a_n)\in R^M$
  • Then $M^{\prime }$ is well-defined (i.e. the definition of $M^{\prime }$ does not depend on the choice of the representatives of the equivalence classes), and called the quotient structure defined by $E$ and denote by $M^{\prime }=\frac{E}{M}$, and $I$ is epimorphism

• (6.21) let $H:M\to M^{\prime }$ be an epimorphism, and $E$ be a binary realtion in $M$ where $(a,b)\in E⟺H(a)=H(b)$, then $E$ is congruence relation, and the quotient structure $\frac{E}{M}$ and $M^{\prime }$ are isomorphic