More generally, a

- Id
∈_{E}*G*

- ∀
*f,g*∈*G*,*g*০*f*∈*G*.

A *permutation* of a set *E *is a bijective
transformation of *E*.

The set ⤹*E *={*f *∈ *E** ^{E}*| Inj

A

*G *⊂
⤹*E* ∧ ∀*f*∈*G*, *f*^{ -1}∈ *G*.

Unlike full transformation monoids and symmetric groups, the concepts of transformation monoid and permutation group make sense independently of the powerset, as sets of transformations satisfying the above first-order stability axioms, ignoring the containing sets

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

Proof of 〈{

IdProof of_{E}∈〈L〉_{{Id, ০}}∴x∈K

(∀f∈〈L〉_{{Id, ০}},∀g∈L,g০f∈〈L〉_{{Id, ০}}∴g(f(x))∈K) ∴K∈Sub_{L}E

TheL⊂ {f∈E| 〈{^{E}x}〉_{L}∈ Sub_{{f}}E} ∈ Sub_{{Id, ০}}E∎^{E}

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

Namely, the concept of monoid is the theory made of

- One type
- 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 more simply 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

An operation • is called *cancellative* if all
transformations defined by currying it
on any side are injective:

∀*x*,*y,z*,
(*x*•*y*=*x*•*z* ⇒*y*=*z* ) ∧
(*x*•*z*=*y*•*z* ⇒*x*=*y* ).

Not all monoids are cancellative. For example the monoid of addition in the set of 3 elements {0,1, several} is not cancellative as 1+several = several+several .

Replacing the presence of *e* in the language by the existence quantifier
on it in the axiom, weakens the concepts of submonoids and morphism of monoid
as follows.

For any function *f* : *M*→ *X* from a monoid (*M*,
*e*, •) to a set with a binary operation (*X*, ▪) preserving
composition (∀*a*,*b*∈*M*, *f*(*a*
• *b*) = *f*(*a*) ▪ *f*(*b*)), Im *f* forms a
monoid where the role of the identity element is played by *f*(*e*).

If the target forms a monoid (*X*,▪, *e*') then (by uniqueness of the identity in Im *f*)

*f*(*e*)=*e'* ⇔ *e'*∈ Im *f* ⇔ Im *f* ∈ Sub_{{e, •}} *X*

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. Categories

3.7. Algebraic terms and term algebras

3.8. Integers and recursion

3.9. Arithmetic with addition