The set π

A

In a permutation group, trajectories are usually called

β_{xβE}
β©{*x*}βͺ_{M}

While the concepts of full transformation monoid and symmetric group depend on the powerset, those of transformation monoid and permutation group do not use it, being expressed as first-order theories.

An element

*y*β’*x* = *e* = *x*β’*z* β *y*
= *y*β’*x*β’*z* = *z*

*x*β’*y* = *y*β’*x* β *x* = *y*β’*x*β’*z*
β *z*β’*x* = *x*β’*z*

β*x*,*y*β*E*,
(*x* *y*)_{E} = (*E* β *z* β¦ (*z* = *x* ? *y* :
(*z* = *y* ? *x* : *z*))) = (*y* *x*)_{E} β π_{E}

Any transposition is involutive (this may give the simplest formal proof that it is a permutation).

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

If a transformation monoid is a group then it is a permutation group (as any inverse of a transformation in the sense of monoids is also its inverse as a function).

A

The

*e*is its own inverse.- If
*x*,*y*have inverses*x*^{-1},*y*^{-1}, then*x*β’*y*has inverse*y*^{-1}β’*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 generated by *A*βͺ-*A* where *A* =
{*x*^{-1}|*x*β*A*}.

Let us qualify an action after this: an

If an *M*-set is both monogenic and generated by the set of its free elements,
then it is regular (there is a free element that generates it).

Proof: as a generator is in the set generated by that of free elements, it
must be in the trajectory of one of them, which is thus also generating.β

(A monogenic action of a monoid may have free elements without being generated by them ;
but if a monogenic action of a group has a free element then all its elements are free).

The trajectory *Y* of any element *x* of an *M*-set *X* is *M*-stable,
and thus an *M*-set. It is generated by *x*, and stays so when replacing the language
*M* by its image as a transformation monoid *N* β *Y ^{Y}*.

Now if

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

As inversion is an anti-morphism,
it switches any action β
of *G* on *X* into a co-action βͺ by
β*x*β*X*, β*g*β*G*, *xβͺg* = *g*^{-1}β
*x*.

If an action of group is monogenic then every element is generating; if it is regular then every
element is regular.

β*P*βπ_{E}, sInv *P* = Inv (*P*
βͺ -*P*) = Inv (*P*) β© Inv(-*P*) β Rel_{E}

β*L*βRel_{E}, β*P*β π_{E}, *L* β sInv
*P* β *P* β Aut_{L} *E*.

Iso(*E*,*F*) = {*f*βMor(*E*,*F*) | β*g*βMor(*F*,*E*),
*g*β*f* = 1_{E} β§ *f*β*g* = 1_{F})}

A *groupoid* is a category where all morphisms are invertible.
Groups are the groupoids with only one object, and the monoid of endomorphisms of any object
of a groupoid is a group.

Generalizing from monoids, the *core* of a category is the groupoid with
the same objects and only accepting isomorphisms as morphisms.

An *automorphism* of an object *E*, is an isomorphism from *E* to itself.
Their set Aut(*E*), core of End(*E*), is a group called the automorphism group of *E*.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory

3.1. Galois connections4. Arithmetic and first-order foundations

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. The category of relations

5. Second-order foundations

6. Foundations of Geometry

Other languages:

FR :
InversibilitΓ© et groupes