A

- Id
∈_{E}*M*

- ∀
*f,g*∈*M*,*g*০*f*∈*M*.

∀*x*∈*E*,
〈{*x*}〉_{L} = {*f*(*x*)|*f*∈〈*L*〉_{{Id,০}}}

∀*X*⊂*E*, 〈*X*〉_{L} =
⋃_{f∈〈L〉{Id,০}} *f*[*X*]
= ⋃_{x∈X} 〈{*x*}〉_{L}

Proof of

IdProof of_{E}∈M∴X⊂K

∀g∈L, ∀y∈K, ∃(f,x)∈M×X,y=f(x) ∧g০f∈M∴g(y) =g০f(x)∈KK∈ Sub_{L}E

TheL⊂ {f∈E|^{E}A∈ Sub_{{f}}E} = End_{{A}}E∈ Sub_{{Id,০}}E^{E}

M⊂ End_{{A}}E

∀f∈M,X⊂A∈ Sub_{{f}}E∴f[X] ⊂A. ∎

〈{*x*}〉_{M} = {*f*(*x*)|*f*∈*M*} ⊂ *E*

End_{Inv(n)L} *E* =
{*g*∈*E ^{E}*| ∀

End

- 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 : ∀
*x*,*x*•*e*=*x*=*e*•*x*

- Associativity : ∀

Both equalities in the last axiom may be considered separately, forming two different concepts

- 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

As the identity axiom ensures the surjectivity of •, every embedding between monoids is injective.

Any {

Similarly it is right cancellative if

If a right identity

An operation is called

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.

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

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 for transformation monoids

This concept will be later generalized to clones of operations with all arities.

A binary operation • in a set *E*, is called
*commutative* when *C*(*E*) = *E*, i.e. *x*,*y*∈*E*, *x*•*y*
= *y*•*x*.

Proof:

In the case of monoids the conditions

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*)

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. Initial and final objects

3.9. Algebraic terms

3.10. Term algebras

3.11. Integers and recursion

3.12. Presburger Arithmetic