∀_{gr}*R*,*S*, *R*;*S* = {(*x*_{0},*y*_{1}) |
(*x*,*y*)∈*R*×*S* ∧ *x*_{1}=*y*_{0}}

∀*R*⊂*E*×*F*, ∀*S*⊂*F*×*G*, *R*;*S* =
{(*x*,*z*)∈*E*×*G* | ∃*y*∈*F*, *xRy* ∧ *ySz*}

= ∐_{x∈E}
*S*^{⋆}*R*⃗(*x*)

(This is the only natural way to extend composition, as the other expression of the composite of functions by their graphs, with ∀

This category faithfully acts by direct images (

∀*A*⊂*E*,
(*S*⚬*R*)^{⋆}*A* = *S*^{⋆}(*R*^{⋆}*A*)

∀*B*⊂*G*, (*R*;*S*)_{⋆}*B* = *R*_{⋆}(*S*_{⋆}*B*)

Thus, coproducts in this category are given by the disjoint unions of objects (which operate as products on the powersets acted on). So, a coproduct of pairwise disjoint sets

By symmetry of duality, this disjoint union also serves as a product.

The eggs of this action are the ({

＊×(*R*^{⋆}*A*) = (＊×*A*);*R*

(*R*_{⋆}*B*)×＊ = *R*;(*B*×＊)

(＊×*A*);*R*;(*B*×＊) = ∅ ⇔
*A* ∩ *R*_{⋆}(*B*) = ∅ ⇔
*B* ∩ *R*^{⋆}(*A*) = ∅

(∁_{F}⚬*R*^{⋆}, ∁_{E}⚬*R*_{⋆}) ∈ Gal(℘(*E*), ℘(*F*))

(*R*^{⋆}, ∁_{E}⚬*R*_{⋆}⚬∁_{F}) ∈
Gal^{+}(℘(*E*), ℘(*F*))

*R*⊂*S* ⇒ *R*;*T* ⊂ *S*;*T*

*R*⊂*S* ⇒ *T*;*R* ⊂ *T*;*S*

(*R*∪*S*);*T* = *R*;*T* ∪ *S*;*T*

*T*;(*R*∪*S*) = *T*;*R* ∪ *T*;*S*

*T*;⋃_{i∈I} *R _{i}* =
⋃

∀*S*⊂*E*×*F*, *S* ⊂ *R*;*S*

∀*S*⊂*F*×*E*, *S* ⊂ *S*;*R*

Thus if

*A* ∈ Sub_{R} *E* ⇔ *R*^{⋆}*A*
⊂ *A* ⇔ (∀(*x*,*y*)∈*R*, *x*∈*A* ⇒ *y*∈*A*) ⇔
∁_{E} *A* ∈ Sub_{tR} *E*

*E* ∋ *z* ↦ (*z*=*x* ? *y* : *z*).

∀*x*,*y*∈*E*,
(*x*,*y*)∈Po(*G*⃗[*K*]) ⇔ (∀*z*∈*K*, *zGx* ⇒
*zGy*) ⇔ *G*⃖(*x*)⊂*G*⃖(*y*)

Now the inclusion order on any

The closure Po(Sub_{R}*E*) of any *R* ⊂ *E*^{2} is called the
preorder on *E* generated by *R*; we shall write it ⌈*R*⌉.

(This does not generalize to structures of other arities).

First proof. (∀*i*∈*I*, *R*^{⋆}*A _{i}*
⊂

Second proof. Use

Proof. First, when *A* is a singleton {*x*}, both definitions coincide :

∀*y*∈*E*, *x*⌈*R*⌉*y* ⇔
(∀*A*∈Sub_{R}*E*, *x*∈*A* ⇒ *y*∈*A*)
⇔ *y*∈〈{*x*}〉_{R}

We conclude by ⌈

From the end of 3.3 we get ∀*A*⊂*E*, 〈*A*〉_{R} =
*A* ∪ *R*^{⋆}〈*A*〉_{R}.

By the faithfulness of the action, it can also be written

⌈*R*⌉ = 𝛿_{E} ∪ *R*⚬⌈*R*⌉

*T* = *R*⚬⌈*R*⌉ = ⌈*R*⌉⚬*R*.

Then, ⌈

*R*⚬⌈*R*⌉ ⊂ ⌈*R*⌉ = ⌈*R*⌉⚬⌈*R*⌉ ∴
⌈*R*⌉⚬*R*⚬⌈*R*⌉⚬*R* ⊂ ⌈*R*⌉⚬⌈*R*⌉⚬*R* = ⌈*R*⌉⚬*R*

This does not depend on

Any well-founded relation is irreflexive : ∀

**Proposition.** The transitive closure *T* of a well-founded relation *R* is well-founded.

∀*A*⊂*E*,
*A* ⊂ *T*^{⋆}*A* ⇒ *P*^{⋆}*A* =
*A*∪ *T*^{⋆}*A* = *R*^{⋆}*P*^{⋆}*A* = ∅ ∎

**Proposition.** If *R* is well-founded then ⌈*R*⌉ is an order, whose strict order coincides with
the transitive closure of *R*.

(The antisymmetry of a well-founded

∀*A*⊂*E*, *A* = ∅ ∨ ∃!:*A*\(*R*^{⋆}*A*)

Given that the corresponding total order ≤ equals ∁

∀*A*⊂*E*, *A* = ∅ ∨ ∃!*x*∈*A*, *A* ⊂ ≤⃗(*x*)

By antisymmetry of ≤, these relations are functional from ℘(

*x* :min *A* ⇔ (*x*∈*A* ∧ *A* ⊂ ≤⃗(*x*))
⇔ (*A*∈ Dom min ∧ *x* = min *A*)

*x* :max *A* ⇔ (*x*∈*A* ∧ *A* ⊂ ≤⃖(*x*))
⇔ (*A*∈ Dom max ∧ *x* = max *A*)

*xSy* ⇔ *y* :min <⃗(*x*) ⇔ ≤⃗(*y*) = <⃗(*x*)

If the order is total then

*xSy* ⇔ *x* :max <⃖(*y*) ⇔ ≤⃖(*x*) = <⃖(*y*)

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. Algebras

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.

6. Foundations of Geometry