Again, we shall usually consider a class of

*s _{E}* :

φ

Algebras form a concrete category with the following sets of morphisms. For any

Mor_{L}(*E*,*F*) = {*f*∈*F ^{E}* |
∀(

Let *L'* be the copy of *L* as a relational language,
where the copy *s'*∈*L'* of each *s*∈*L* has
increased arity *n _{s'}* =

* ^{L'}E* ⇌ ∐

∐

All concepts defined for relational systems are equally applicable through this canonical bijection.

*Functional*or a*partial algebra*if**E**is functional (graph of a function from a subset of);^{L}E*Serial*if Dom**E**=;^{L}E*Algebraic*if a serial partial algebra, i.e. essentially an algebra (*E*, φ_{E}) by**E**= Gr φ_{E};*Injective*if^{t}**E**is functional ;*Surjective*if Im**E**=*E*.

∀*f*∈*F ^{E}*, (∀(

Mor_{L}(*E*,*F*) =
{*f*∈*F ^{E}* | Im

Mor

- If
*F*is functional then*E*is functional; -
If
*F*is injective then*E*is injective.

- If
*E*is serial then*F*is serial; -
If
*E*is surjective then*F*is surjective.

We shall say that

Sub_{L} *E* = {*A*⊂*E* | **E**^{⋆}(* ^{L}A*)
⊂

If

If

Conversely, any algebraic subset of a partial algebra is stable.

If a formula of the form (∀(variables), formula without binder) is true in an

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

Other way:
**E**^{⋆}(* ^{L}*⋂

〈*A*〉_{L} =
{*x*∈*E* | ∀*B*∈ Sub_{L}*E*, *A*⊂*B*
⇒ *x*∈*B*} =
⋂{*B*∈ Sub_{L}*E* | *A*⊂*B*} ∈ Sub_{L}*E*

∀*X*⊂*E*, ∀*Y*⊂Sub_{L}*E*, *X* ⊂ ⋂*Y*
⇔ (∀*B*∈*Y*, *X*⊂*B*) ⇔ *Y* ⊂ {*B*∈ Sub_{L}*E* | *X*⊂*B*}.

**Stability of equalizers.** The *equalizer* Eq(*f*, *g*) = {*x*∈*E* | *f*(*x*) = *g*(*x*)} of any two *L*-morphisms *f*,*g*∈Mor* _{L}*(

∀(

Proof when

Min_{L}*E* = 〈∅〉_{L,E}
= ⋂ Sub_{L} *E* ∈ Sub_{L} *E*.

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

For any *L*-system *E* and any *A*⊂*E*,

- ∀
*B*∈ Sub_{L}*E*,*A*∩*B*∈ Sub_{L}*A* - Min
_{L}*A*⊂ Min_{L}*E* *A*is minimal ⇒*A*⊂ Min_{L}*E**A*∈ Sub_{L}*E*⇒**E**^{⋆}() ∈ Sub^{L}A_{L}*E*∩ ℘(*A*) = Sub_{L}*A**A*∈ Sub_{L}*E*⇒ Min_{L}*A*= Min_{L}*E**A*= Min_{L}*E*⇔ (*A*is minimal ∧*A*∈ Sub_{L}*E*) ⇒**E**^{⋆}() =^{L}A*A*

In particular, any minimal system is surjective.- 〈
*A*〉_{L}= Min_{L∪A}*E*where*A*is seen as a set of constants. *A*∪**E**^{⋆}() ⊂ 〈^{L}A*A*〉_{L}=*A*∪**E**^{⋆}(〈^{L}*A*〉_{L})**E**_{⋆}*A*⊂⇒ Min^{L}A_{L}*A*=*A*∩ Min_{L}*E*

Proof. 〈∁_{E} Im *g*〉_{{g⚬f}} = *A* =
(∁_{E} Im *g*) ∪ *g*[*f*[*A*]] ⇒ ∁_{E}
*A* = *g*[∁_{F} *f*[*A*]] ⇒ (*E* ∋ *x* ↦
*x*∈*A* ? *f*(*x*) : *g*^{-1}(*x*)) : *E* ↔ *F* ∎

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory

- 3.1. Galois
connections

3.2. Relational systems and concrete categories

3.3.

3.4. Special morphisms

3.5. Monoids and categories

3.6. Actions of monoids and categories

3.7. Invertibility and groups

3.8. Properties in categories

3.9. Initial and final objects

3.10. Products of systems

3.11. Basis

3.12. The category of relations

5. Second-order foundations

6. Foundations of Geometry