Again, let us assume a fixed class of

φ_{E} = ∐_{s∈L}
*s _{E}* = ((

**Morphisms of algebras. **For any *L*-algebras *E*,
*F*,

Such categories can be seen as particular categories of relational systems, as follows.

Let the relational languageL'be a copy ofLwhere to eachs∈Lcorrespondss'∈L'with increased arityn=_{s'}n+1, so that_{s}also expressible as the set of triples ( L'⋆E≡ ∐_{s∈L}E^{ns}×E≡ (L⋆E)×Es,x,y) such thats∈L,x∈Eand^{ns}y∈E.

Eachn-ary operation_{s}sdefines an_{E}n-ary relation_{s'}s'Gr_{E}≡s. These are packed as an_{E}L'-structure

The resulting condition for anE= Gr φ_{E}≡ ∐_{s∈L}s'._{E}f∈Fto be a morphism is equivalent :^{E}(∀( x,y)∈E, (f(_{L}x),f(y))∈F) ⇔ (∀x∈L⋆E, φ_{F}(f(_{L}x))=f(φ_{E}(x))).

**Subalgebras**. A subset *A*⊂*E* of an *L*-algebra
*E* will be called an *L-subalgebra* of *E*, if
φ_{E}[*L*⋆*A*]⊂*A*.

Then the restriction φ_{A} of φ_{E}
to *L*⋆*A* gives it a structure of *L*-algebra.

The set of all *L*-subalgebras of *E*
will be denoted Sub_{L} *E*. It is nonempty
as *E* ∈ Sub_{L} *E*.

**Images of algebras**. For any two *L*-algebras *E*,*F*,
∀*f* ∈Mor_{L}(*E*,*F*), Im *f* ∈ Sub_{L}*F*.

∀(Thus trying to exend this result to algebras with infinitary operations, would require the axiom of choice, otherwise it anyway still holds for injective morphisms.s,y)∈L⋆Imf, ∃x∈E^{ns},f০x=y∴s(_{F}y) =f(s(_{E}x)) ∈ Imf∎

Let us generalize the concept of algebra, to any *L'*-systems
(*E*,**E**), where **E** ⊂ (*L*⋆*E*)×*E*
needs not be functional. They form the same kind of categories previously
defined, with a different notation (through the canonical bijection depending on the choice of
distinguished argument) by which more concepts can be introduced.

*A* ∈ Sub_{L} *E* ⇔ (**E**_{*}(*L*⋆*A*)
⊂*A*) ⇔ (∀(*s*,*x*,*y*)∈**E**,
Im *x*⊂*A* ⇒ *y*∈*A*).

**Preimages of stable subsets.** ∀*f*∈Mor* _{L}*(

∀(Proof fors,x)∈L⋆A,f০x∈B^{ns}∴f(s(_{E}x)) =s(_{F}f০x) ∈B∴s(_{E}x)∈A.

∀(x,y)∈E, (f(_{L}x),f(y))∈F∴ (x∈L⋆A⇒f(_{L}x)∈L⋆B⇒f(y)∈B⇒y∈A).∎

∀*f*,*g*∈Mor* _{L}*(

**Intersections of stable subsets.** ∀*X* ⊂ Sub_{L}*E,*
∩*X* ∈ Sub_{L} *E* where ∩*X *≝
{*x*∈*E*|∀*B*∈*X*, *x*∈*B*}.

Other way:
**E**_{*}(*L*⋆∩*X*) =
**E**_{*}(∩_{B∈X} *L*⋆*B*)
⊂∩_{B∈X}
**E**_{*}(*L*⋆*B*) ⊂∩*X*.

**Subalgebra generated by a subset.** ∀*A* ⊂ *E*, the
*L-subalgebra of E generated by A*, written
〈*A*〉_{L,E} or more simply 〈*A*〉_{L},
is the smallest *L*-subalgebra of *E* including *A*:

For fixed

We say that

An *L*-algebra *E* is * minimal* when *E* = Min_{L}
*E*, or equivalently Sub_{L}*E*
= {*E*}.

- ∀
*B*⊂*A*,*B*∈Sub_{L}*E*⇔*B*∈Sub_{L}*A* - Min
_{L}*A*= Min_{L}*E* *A*=Min_{L}*E*⇔*A*is minimal.

Min(Among subsets of_{L}E⊂ Min_{L}Abecause Sub_{L}A⊂ Sub_{L}E.

Min_{L}A⊂ Min_{L}Ebecause ∀B∈Sub_{L}E,A⋂B∈ Sub_{L}A. ∎

We can redefine generated subalgebras in terms of minimal subalgebra with a different language: 〈

**Injective, surjective algebras.** An *L*-algebra (*E*,φ_{E})
will be called injective if φ_{E} is injective, and surjective if
Im φ_{E} = *E*.

**Proposition.** For any *L*-algebras *E*, *F*,

- ∀
*A*⊂*E*, Im φ_{E}⊂*A*⇒*A*∈Sub_{L}*E*. - Any minimal
*L*-algebra is surjective. - Min
_{L}*E*= φ_{E}[*L*⋆Min_{L}*E*] ⊂ Im φ_{E} - ∀
*A*⊂*E*, 〈*A*〉_{L}=*A*∪φ_{E}[*L*⋆〈*A*〉_{L}] ⊂*A*∪Im φ_{E}. -
∀
*f*∈Mor_{L}(*E*,*F*),*f*[Min_{L}*E*] = Min_{L}*F*; more generally ∀*A*⊂*E*,*f*[〈*A*〉_{L}] = 〈*f*[*A*]〉_{L}

- φ
_{E}[*L*⋆*A*] ⊂ Im φ_{E}⊂*A*⇒*A*∈Sub_{L}*E* - Im φ
_{E}∈ Sub_{L}*E* - Min
_{L}*E*is surjective *A*∪φ_{E}[*L*⋆〈*A*〉_{L}] ∈ Sub_{L}〈*A*〉_{L}- ∀
*B*∈ Sub_{L}*F*,*f**(*B*)∈Sub_{L}*E*∴ Min_{L}*E*⊂*f**(*B*) ∴*f*[Min_{L}*E*]⊂*B*.∎

**Injectivity lemma.** If *E* is a surjective algebra and
*F* is an injective one then ∀*f* ∈Mor_{L}(*E*,*F*),

*A*= {*x*∈*E*| ∀*y*∈*E*,*f*(*x*) =*f*(*y*) ⇒*x*=*y*} ∈ Sub_{L}*E*.- For each uniqueness quantifier
*Q*(either ∃! or !),*B*= {*y*∈*F*|*Q**x*∈*E*,*y*=*f*(*x*)} ∈ Sub_{L}*F*

- ∀(
*s*,*x*)∈*L*⋆*A*, ∀*y*∈*E*,

*f*(*s*(_{E}*x*)) =*f*(*y*) ⇒ (∃(*t*,*z*)∈φ_{E}^{•}(*y*),*s*(_{F}*f*০*x*) =*f*(*s*(_{E}*x*)) =*f*(*y*) =*f*(*t*(_{E}*z*)) =*t*(_{F}*f*০*z*) ∴ (*s*=*t*∧*f*০*x*=*f*০*z*) ∴*x*=*z*) ⇒*s*(_{E}*x*)=*y*. - As φ
_{F}is injective, ∀*y*∈φ_{F}[*L*⋆*B*], ∃!: φ_{F}^{•}(*y*) ⊂*L*⋆*B*∴*Q**z*∈*L*⋆*E*, φ_{F}(*f*(_{L}*z*)) =*y*.

As φ_{F}০*f*=_{L}*f*০φ_{E}and φ_{E}is surjective, we conclude*Q**x*∈*E*,*y*=*f*(*x*). ∎

**Schröder–Bernstein theorem**.
If there exist injections *f*: *E* → *F* and
*g*: *F*→ *E* then there exists a bijection between *E* and *F*.

Then a bijection from

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of relational systems and concrete categories4. Model Theory

3.2.Algebras

3.3. Special morphisms

3.4. Monoids

3.5. Actions of monoids

3.6. Invertibility and groups

3.7. Categories

3.8. Algebraic terms and term algebras

3.9. Integers and recursion

3.10. Arithmetic with addition