A

- Id
∈_{E}*M*

- ∀
*f,g*∈*M*,*g*⚬*f*∈*M*.

- Two operation symbols
- a constant symbol
*e*of "identity" - a binary operation • of "composition"

- a constant symbol
- Axioms
- Associativity : ∀
*x*,*y,z**, x*•(*y*•*z*) = (*x*•*y*)•*z*so that either term can be written*x*•*y*•*z* - Identity axiom : ∀
*x*,*x*•*e*=*x*=*e*•*x*

- Associativity : ∀

- A
**left identity**of a binary operation • is an element*e*such that ∀*x*,*e*•*x*=*x*; - A
**right identity**of • is an element*e'*such that ∀*x*,*x*•*e'*=*x*.

From any associative operation on a set *E* we can form a monoid by adding the
identity *e* as an extra element, *E'* = *E* ⊔ {*e*}*E* loses its status of identity in *E'*.

∀*y,z*, *x*•*y* = *x*•*z* ⇒ *y* = *z*.

∀*y,z*, *y*•*x* = *z*•*x* ⇒ *y* = *z*.

For example the monoid of addition in {0,1, several} is not cancellative as 1+several = several+several.

Any submonoid of a cancellative monoid is cancellative.

The existence of a left cancellative element implies the uniqueness of a right identity :

*a*•*e'* = *a* = *a*•*e* ⇒ *e'* = *e*.

For any monoid (

- its image is a monoid (
*A*,*a*,▪) where*A*= Im*f*and*a*=*f*(*e*) *f*is a morphism of monoid from*M*to this monoid*A*.

*f* ∈ Mor_{{e}}(*M*,*X*) ⇔
*a* = *e'* ⇔
*e'* ∈ *A* ⇔ *A* ∈ Sub_{{e, ▪}} *X*

**Anti-morphisms**. The *opposite* of a monoid is the monoid
with the same base set but where composition is replaced by its transpose. An
*anti-morphism* from (*M*,*e*,•) to (*X*,*e'*,▪) is a morphism
*f* from one
monoid to the opposite of the other (or equivalently vice-versa):

*f*(*e*) = *e'*

∀*a*,*b*∈*M*, *f*(*a*•*b*) =
*f*(*b*)▪*f*(*a*)

*C*(*A*) = {*x*∈*E*|∀*y*∈*A*,
*x*•*y *= *y*•*x*}.

∀*A*,*B*⊂*E*,
*B*⊂*C*(*A*) ⇔ *A*⊂*C*(*B*).

If • is associative then ∀*A*⊂*E*, *C*(*A*)
∈ Sub_{{•}}*F*.

Proof: ∀*x,y*∈*C*(*A*),
(∀*z*∈*A*, *x*•*y*•*z* = *x*•*z*•*y* =
*z*•*x*•*y*) ∴ *x*•*y*∈*C*(*A*).

This can be understood as an intersection of submonoids in the case of a transformation monoid

∀*A*⊂*M*,
*C _{M}*(

A binary operation • in a set *E*, is called
*commutative* when • = ^{t}•,
i.e. *C*(*E*) = *E*.

If *A*⊂*C*(*A*) and 〈*A*〉_{{•}} = *E* then • is commutative.

Proof: *A*⊂*C*(*A*)∈ Sub_{{•}}*F*
⇒ *E* = *C*(*A*)
⇒ *A*⊂*C*(*E*) ∈ Sub_{{•}}*F* ⇒
*C*(*E*) = *E*.∎

- the class of its "objects" ; the category is
*small*if this class is a set; - to any objects
*E*,*F*is given a set Mor(*E*,*F*) of «morphisms from*E*to*F*»; these are usually regarded as pairwise disjoint; - to any object
*E*is given a constant symbol 1_{E}∈ Mor(*E*,*E*); - to any triple of objects
*E*,*F*,*G*is given a "composition" operation (abusively all denoted by the same symbol)

∘ :
Mor(*F*,*G*)×Mor(*E*,*F*) → Mor(*E*,*G*)

- ∀
_{C}*E*,*F*, ∀*x*∈Mor(*E*,*F*),*x*∘1_{E}=*x*= 1_{F}∘*x* - ∀
_{C}*E,F,G,H,*∀*x*∈Mor(*E*,*F*), ∀*y*∈Mor(*F*,*G*),∀*z*∈Mor(*G*,*H*), (*z*∘*y*)∘*x*=*z*∘(*y*∘*x*) ∈ Mor(*E*,*H*)

The concept of *opposite category* is defined similarly to opposite monoids,
as re-reading the category with sides reversed, the Mor(*E*,*F*) of the one serving as
the Mor(*F*,*E*) of the other.

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.

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

5. Second-order foundations

6. Foundations of Geometry

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Monoïdes et catégories