The set ⤹

A

While the concepts of full transformation monoid and symmetric group depend on the powerset, those of transformation monoid and permutation group can be defined independently of it, as first-order theories with 2 types.

Trajectories are usually calledFor any transformation monoid or action of a monoid

Seeing

∀*z,t*∈*M*, *y*•*z* = *t* ⇒
*x*•*t* = *z*

*x*•*y* =
*e* ⇒ ∀*z*∈*M*, *z*•*x*•*y* = *z*

∀*z,t*∈*M*, (*y*•*z* = *y*•*t* ∧ *x*•*y* = *e*) ⇒
(*z* = *x*•*y*•*z* = *x*•*y*•*t*
= *t*)

*y*•*x* = *e* = *x*•*z* ⇒ *y*
= *y*•*x*•*z* = *z*

If

*x*•*y* = *y*•*x* ⇔ *x* = *y*•*x*•*z*
⇔ *z*•*x* = *x*•*z*

- The axiom that all elements are invertible, or
- The function symbol
^{-1}of inversion, with the axiom ∀*x*,*x*•*x*^{-1}=*x*^{-1}•*x*=*e*

Permutation groups are the transformation monoids which are groups (in the first above sense).

A

The set of invertible elements in any monoid

- If
*x*,*y*have inverses*x*^{-1},*y*^{-1}, then*x*•*y*has inverse*y*^{-1}•*x*^{-1}. - Any inverse
*x*^{-1}is invertible, with (*x*^{-1})^{-1}=*x*(inversion is an involutive transformation of any group).

Between groups, a

In a group, the subgroup generated by a subset *A*, coincides with the
submonoid *G* generated by *A*∪-*A* where -*A* =
{*x*^{-1}|*x*∈*A*}. (To check that *G* is stable by
inversion, notice that the definition of *G* is
stable by inversion, which is involutive, thus *G* = *G*

Now this can qualify actions (

Let us call it

Proof: a generator being generated by the set of free elements, must be in the trajectory of one of them, which is thus also generating. (On the other hand, a monogenic action may have free elements without being free).

An *action* of a group *G* on a set *X*, is equivalently
an action of monoid, or a group morphism from *G* to the symmetric group of *X*.

By the representation theorem, any group is isomorphic to a permutation group, namely the group of automorphisms of an algebra.

If an action of group is monogenic then every element is generating ; if it is free then all elements are free, so that all parts of its partition into orbits are regular.

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. Algebraic terms

3.9. Term algebras

3.10. Integers and recursion

3.11. Presburger Arithmetic