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 languageEvery injective morphismL'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

E= Gr φ_{E}≡ ∐_{s∈L}s'._{E}

The conditions for anf∈Fto be a morphism are equivalent :^{E}(∀( x,y)∈E, (f(_{L}x),f(y))∈F) ⇔ (∀x∈L⋆E, φ_{F}(f(_{L}x))=f(φ_{E}(x))).

∀(*s*,*x*,*y*)∈*L'*⋆*E*, *f*(*y*)
= *s _{F}*(

Bijective morphisms of algebras are isomorphisms. This can be deduced from the fact they are embeddings, or by

φ_{E}০*f _{L}*

**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*.

For any formula of the form (∀(variables), some formula without
any binder), its truth in *E* implies its truth in each *A*∈Sub_{L}
*E*.

**Images of algebras**.* f* ∈Mor* _{L}*(

∀(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.

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

∀(

Proof for

∀(

**Proposition.** For any *L'*-system *E* and any
*L*-algebra *F*,

∀*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

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

**Proposition.** For any *L*-algebra *E*,

- ∀
*A*⊂*E*, Imφ_{E}⊂*A*⇒*A*∈Sub_{L}*E*. In particular, Im φ_{E}∈Sub_{L}*E*. -
∀
*A*⊂*E*,*A*⊂〈*A*〉_{L}⊂*A*∪ Imφ_{E}. In particular, Min_{L}*E*⊂ Imφ_{E} -
∀
*A*⊂*E*, 〈*A*〉_{L}=*E*⇒*A*∪Imφ_{E}=*E*. -
∀
*f*∈Mor_{L}(*E*,*F*),*f*[Min_{L}*E*] = Min_{L}*F*(images of minimal algebras by morphisms are minimal)

φ_{E}[L⋆A] ⊂ Imφ_{E}⊂A

Imφ_{E}⊂A∪Imφ_{E}∴A∪Imφ_{E}∈Sub_{L}E

∀B∈ Sub_{L}F,f*(B)∈Sub_{L}E∴ Min_{L}E⊂f*(B) ∴f[Min_{L}E]⊂B.∎

**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*.

Any minimal *L*-algebra is surjective. Thus, the minimal
sub-algebra of any algebra is also a surjective algebra.

**Proposition.** 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*.

Other view : under the same assumptions, for each uniqueness quantifier *Q* (either ∃! or !),

*B* = {*y*∈*F* | *Q**x*∈*E*, *y* =
*f*(*x*)} ∈ Sub_{L}*F*

As φ

More texts on algebra

Back to homepage : 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. Special morphisms

3.3.Algebras

3.4. Algebraic terms and term algebras

3.5. Integers and recursion

3.6. Arithmetic with addition

4.1. Finiteness and countability

4.2. The Completeness Theorem

4.3. Infinity and the axiom of choice

4.4. Non-standard models of Arithmetic

4.5. How theories develop

4.6. The Incompleteness Theorem