*E*=^{n}*E*×...×*E*=*E*^{Vn}, set of all*n*-tuples of elements of*E*(*n*-ary product or exponentiation).- Rel
_{E}^{(n)}= ℘(*E*), set of all^{n}*n*-ary relations in*E* - Op
_{E}^{(n)}=*E*, set of all^{En}*n*-ary operations in*E*.

^{L}*E* = ∐_{s∈L}
*E ^{ns}*

∏_{s∈L} ℘(*E ^{ns}*)
≅ ℘(

The case of an *algebraic language*, whose symbols aim to represent operations,
will be studied in 3.2.

*s _{E}* =
{

Im

∀

*f* ∈ Mor_{L}(*E*,*F*) ⇔
∀(*s*,*x*)∈**E**, (*r*,*f*০*x*)∈**F**

⇔ (∀*s*∈*L*,∀*x*∈*E*^{ns},
*s*(*x*) ⇒ *s*(*f*০*x*))

⇔
* ^{L}f*[

- a class of sets called
*objects*; - a class of functions called
*morphisms*; then for any objects*E*,*F*, we define

Mor(*E*,*F*) = {*f*∈*F*|^{E}*f*is a morphism}

- Every morphism belongs to some Mor(
*E*,*F*), i.e. its domain is an object and its image is included in an object (in practice, images of morphisms will be objects too); - For any object
*E*, Id_{E}∈ Mor(*E*,*E*) ; - Any composite of morphisms is a morphism: for any 3 objects
*E*,*F*,*G*, ∀*f*∈ Mor(*E*,*F*), ∀*g*∈Mor(*F*,*G*),*g*০*f*∈ Mor(*E*,*G*).

A relational symbol

A category is *small* if its class of objects
is a set.

Indeed, for any

- Substituting arguments of a
*s*∈*L*by a map σ to*n*' other variables (∀*E*,∀*x*∈*E*^{n'},*s'*(*x*)⇔*s*(*x*০σ)), works :*s'*(*x*) ⇒*s*(*x*০σ) ⇒*s*(*f*০*x*০σ) ⇒*s'*(*f*০*x*). - ∀
*s*,*s*'∈*L*,*n*=_{s}*n*⇒ ∀_{s'}*x*∈*E*^{ns}, (*s*(*x*)∧*s'*(*x*)) ⇒ (*s*(*f*০*x*)∧*s'*(*f*০*x*)) - ∀
*s*,*s*'∈*L*,*n*=_{s}*n*⇒ ∀_{s'}*x*∈*E*^{ns}, (*s*(*x*)∨*s'*(*x*)) ⇒ (*s*(*f*০*x*)∨*s'*(*f*০*x*)) - For 0 and 1 it is trivial

- ∀
*x*,*y*∈*E*,*x*=*y*⇒*f*(*x*)=*f*(*y*) - ∀
*x*∈*E*^{ns},(∃*y*∈*E*,*s*(*x*,*y*)) ⇒ (∃*z*=*f*(*y*)∈*F*,*s*(*f*০*x*,*z*))

The above cases of 0, 1, ∨ and ∧ are mere particular cases (the nullary and binary cases) of the following:

- Any union of a family of preserved structures in a concrete category is a preserved structure.
- Any intersection of a family of preserved structures is also a preserved structure.

**Proposition**. In any concrete category, for any choice of *n*-tuple *t*
of elements of some object *K*, the relation *s* defined in each object *E* as
*s _{E}* = {

From these definitions it might happen between objects

In a small concrete category, the preserved families of relations are precisely all choices of
unions of those : each preserved *s* equals the union of those with *t*
ranging over *s* (with *K* ranging over all objects).

- A tuple (or family) of functions (
*f*)_{t}_{t∈τ}, where ∀*t*∈τ,*f*:_{t}*E*→_{t}*F*where_{t}*E*⊂_{t}*E*,*F*⊂_{t}*F*are the interpretations of type*t*in*E*and*F* - A function
*f*:*E*→*F*that is a τ-morphism seeing τ as a list of unary relation symbols (like for the use of classes as notions in set theory), i.e. such that*h*০_{F}*f*=*h*where_{E}*h*:_{E}*E*→τ,*h*:_{F}*F*→τ are the functions giving the type of each element.

Again, we shall usually consider a class of

*s _{E}* :

φ

These form a concrete category with the following concept of morphism.

**Morphisms of algebras. **For any *L*-algebras *E*,
*F*,

Such categories can be seen as particular categories of relational systems, as follows.

Let the relational languageL'be a copy ofLwhere the copys'∈L'of eachs∈Lhas increased arityn=_{s'}n+1, so that_{s}Each ≡ ∐^{L'}E_{s∈L}E^{ns}×E≡ ()×^{L}EE≡ {(s,x,y) |s∈L∧x∈E∧^{ns}y∈E}.n-ary operation_{s}sdefines an_{E}n-ary relation_{s'}s'Gr_{E}≡s. These are packed as an_{E}L'-structure

The resulting condition for anE= Gr φ_{E}≡ ∐_{s∈L}s'._{E}f∈Fto be a morphism is equivalent :^{E}(∀( x,y)∈E, ((^{L}fx),f(y))∈F) ⇔ (∀x∈, φ^{L}E_{F}((^{L}fx))=f(φ_{E}(x))).

**Subalgebras**. A subset *A*⊂*E* of an *L*-algebra *E* is
called *stable by L* or an *L-subalgebra* of *E*, if
φ_{E}[* ^{L}A*]⊂

**Images of algebras**. For any two *L*-algebras *E*,*F*,
∀*f* ∈Mor_{L}(*E*,*F*), Im *f* ∈ Sub_{L}*F*.

*A* ∈ Sub_{L} *E* ⇔ (**E**_{*}(* ^{L}A*)
⊂

**Preimages of stable subsets.** ∀*f*∈Mor_{L}(*E*,*F*),
∀*B*∈Sub_{L}*F*, *f* *(*B*) ∈ Sub_{L}
*E*.

For

For

**Proposition.** For any *L'*-system *E* and any
*L*-algebra *F*,

∀*f*,*g*∈Mor* _{L}*(

**Intersections of stable subsets.** ∀*X* ⊂ Sub_{L}*E,*
∩*X* ∈ Sub_{L} *E* where ∩*X *≝
{*x*∈*E*|∀*B*∈*X*, *x*∈*B*}.

Other way:
**E**_{*}(* ^{L}*∩

〈*A*〉_{L} =
{*x*∈*E* | ∀*B*∈Sub_{L}*E*, *A*⊂*B*
⇒ *x*∈*B*} =
∩{*B*∈Sub_{L}*E* | *A*⊂*B*} ∈ Sub_{L}*E*

∀

We say that

Min

An *L*-algebra *E* is * minimal* when *E* = Min_{L}
*E*, or equivalently Sub_{L}*E*
= {*E*}.

- ∀
*B*⊂*A*,*B*∈Sub_{L}*E*⇔*B*∈Sub_{L}*A* - Min
_{L}*A*= Min_{L}*E* *A*= Min_{L}*E*⇔*A*is minimal.

Min

Among subsets of *E*, other minimal *L'*-systems are included
in Min_{L} *E* but are not stable.

The stable subset generated by *A* is the minimal one for the extended language
with *A* seen as a set of constants:
〈*A*〉_{L,E}= Min_{L∪A} *E*.

**Proposition.** For any *L*-algebras *E*, *F*,

- ∀
*A*⊂*E*, Im φ_{E}⊂*A*⇒*A*∈Sub_{L}*E*. - Any minimal
*L*-algebra is surjective. - Min
_{L}*E*= φ_{E}[Min^{L}_{L}*E*] ⊂ Im φ_{E} - ∀
*A*⊂*E*, ⋃_{x∈A}〈{*x*}〉_{L}⊂ 〈*A*〉_{L}=*A*∪φ_{E}[〈^{L}*A*〉_{L}] ⊂*A*∪Im φ_{E} -
∀
*f*∈Mor_{L}(*E*,*F*),*f*[Min_{L}*E*] = Min_{L}*F*∧ ∀*A*⊂*E*,*f*[〈*A*〉_{L}] = 〈*f*[*A*]〉_{L}

- φ
_{E}[] ⊂ Im φ^{L}A_{E}⊂*A*⇒*A*∈Sub_{L}*E* - Im φ
_{E}∈ Sub_{L}*E* - Min
_{L}*E*is surjective *A*∪φ_{E}[〈^{L}*A*〉_{L}] ∈ Sub_{L}〈*A*〉_{L}- ∀
*B*∈ Sub_{L}*F*,*f**(*B*)∈Sub_{L}*E*∴ Min_{L}*E*⊂*f**(*B*) ∴*f*[Min_{L}*E*]⊂*B*.∎

**Injectivity lemma.** If *E* is a surjective algebra and
*F* is an injective one then ∀*f* ∈Mor_{L}(*E*,*F*),

*A*= {*x*∈*E*| ∀*y*∈*E*,*f*(*x*) =*f*(*y*) ⇒*x*=*y*} ∈ Sub_{L}*E*.- For each uniqueness quantifier
*Q*(either ∃! or !),*B*= {*y*∈*F*|*Q**x*∈*E*,*y*=*f*(*x*)} ∈ Sub_{L}*F*

- ∀(
*s*,*x*)∈, ∀^{L}A*y*∈*E*,*f*(*s*(_{E}*x*)) =*f*(*y*) ⇒ (∃(*t*,*z*)∈φ_{E}^{•}(*y*),*s*(_{F}*f*০*x*) =*f*(*s*(_{E}*x*)) =*f*(*y*) =*f*(*t*(_{E}*z*)) =*t*(_{F}*f*০*z*) ∴ (*s*=*t*∧*f*০*x*=*f*০*z*) ∴*x*=*z*) ⇒*s*(_{E}*x*) =*y*. - As φ
_{F}is injective, ∀*y*∈φ_{F}[], ∃!: φ^{L}B_{F}^{•}(*y*) ⊂∴^{L}B*Q z*∈, φ^{L}E_{F}((^{L}f*z*)) =*y*.

As φ_{F}০=^{L}f*f*০φ_{E}and φ_{E}is surjective, we conclude*Qx*∈*E*,*y*=*f*(*x*). ∎

**Schröder–Bernstein theorem**.
If there exist injections *f*: *E* → *F* and
*g*: *F*→ *E* then there exists a bijection between *E* and *F*.

Then a bijection from

If

The role it gives to

If

Given an algebraic language *L*, an equivalence relation *R* on *E* is said to be compatible with an
*L*'-structure **E** if the quotient structure is a functional graph. If **E** is an
algebra structure then Dom(**E**/*R*) = * ^{L}*(

For any

(

(

∀*x*∈*E ^{nr}*,

Injectivity is usually added to the definition of the concept of embedding, as it means strongly preserving the equality relation. Things can come down to this case by replacing equality in the concept of injectivity by a properly defined equivalence relation, or replacing systems by their quotient by this relation, where the canonical surjections would be non-injective embeddings.

**Isomorphism**. Between objects *E*
and *F* of a concrete category, an *isomorphism* is a bijective morphism
(*f* ∈Mor(*E*,*F*) ∧ *f* : *E* ↔ *F*)
whose inverse is a morphism (*f*^{ -1}∈Mor(*F*,*E*)). In
the case of relational systems, isomophisms are the bijective embeddings;
injective embeddings are isomorphisms to their images.

Two objects *E*, *F* of a category are said to be *isomorphic* (to each other) if
there exists an isomorphism between them. This is an equivalence predicate, i.e. it works as an equivalence relation on the class of
objects in this category.

The *isomorphism class* of an object in a category, is the class of all objects
which are isomorphic to it. Then an isomorphism class of objects in a category, is a class
of objects which is the isomorphism class of some object in it (independently of the choice).

- Id
_{A}∈Mor(*A*,*E*), therefore Mor(*A*,*N*) ⊂ {*g*_{|A}|*g*∈Mor(*E*,*N*)} - For any system
*N*, Mor(*N*,*A*) = {*h*∈Mor(*N*,*E*) | Im*h*⊂*A*} - An
*f*∈Mor(*N*,*E*) is an isomorphism to*A*if and only if it is an injective embedding to*E*and Im*f*=*A*.

∀(*s*,*x*,*y*)∈* ^{L'}E*,

Bijective morphisms of algebras are isomorphisms. This can be deduced from the fact they are embeddings, or by

(* ^{L}f*)

Thus, they also preserve invariant structures whose formula may use symbols of

An

Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphism

The most usual practice of mathematics ignores the diversity of elementarily equivalent but non-isomorphic systems, as well as non-surjective elementary embeddings. However, they exist and play a special role in the foundations of mathematics, as we shall see with Skolem's paradox and non-standard models of arithmetic.

An

Automorphism ⇔ (Endomorphism ∧ Isomorphism)

An endomorphismImfis also invariant (defined by ∃y∈E,f(y)=x)

∀x∈E,x∈Imf⇔f(x)∈ Imf

Imf=E. ∎

∀*M*⊂*E ^{E}*, ∀

Inv =
∏_{n∈ℕ} Inv^{(n)} : ℘(*E ^{E}*) →
∏

A

- Id
∈_{E}*M*

- ∀
*f,g*∈*M*,*g*০*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 : ∀
*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*)

A

- ∀
*x*∈*X*,*e*⋅*x*=*x*; - ∀
*a*,*b*∈*M*, ∀*x*∈*X*, (*a*•*b*)⋅*x*=*a*⋅(*b*⋅*x*).

In curried view, a left action of

- ∀
*x*∈*X*,*x*⋅*e*=*x*; - ∀
*a*,*b*∈*M*, ∀*x*∈*X*, (*x*⋅*a*)⋅*b*=*x*⋅(*a*•*b*)

The commutation of 2 submonoids of

∀*x*∈*X*,
(*a*⋅*x*)⋅*b* = *a*⋅(*x*⋅*b*)

∀*a*,*b*∈*M*, (∀*x*∈*X*,
*a*·*x* = *b*·*x*) ⇒ *a*=*b*

An element

(∃*x*∈*X*,
∀*a*≠*b*∈*M*, *a*·*x* ≠ *b*·*x*)
⇒ (∀*a*≠*b*∈*M*, ∃*x*∈*X*,
*a*·*x* ≠ *b*·*x*)

*h _{x}*(

∀*g*∈*X ^{M}*, ∀

(Id_{M} ∈
Mor_{M}(*M*,*M*) ∧ Id_{M}(*e*)=*e*)
⇒ *h _{e}* = Id

∀*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

Forgetting

Now if

The proof is easy and left as an exercise.

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*.

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.

∀*P*⊂⤹*E*, sInv *P* = Inv (*P*
∪ -*P*) = Inv (*P*) ∩ Inv(-*P*) ⊂ Rel_{E}

∀*L*⊂Rel_{E}, ∀*P*⊂ ⤹*E*, *L* ⊂ sInv
*P* ⇔ *P* ⊂ Aut_{L} *E*.

- A class of "objects" of that category (which need not be sets);
the category is
*small*if this class is a set; - to any objects
*A*,*B*is given a set Mor(*A*,*B*) of «morphisms from*A*to*B*»; these are usually regarded as pairwise disjoint; - to any object
*A*is given 1_{A}∈ Mor(*A*,*A*); - to any 3 objects
*A*,*B*,*C*is given a composition operation we shall abusively denote by the same symbol • : Mor(*B*,*C*)×Mor(*A*,*B*)→Mor(*A*,*C*) ;

- For any objects
*A*,*B*, ∀*x*∈Mor(*A*,*B*),*x*•1_{A}=*x*= 1_{B}•*x* - For any objects
*A,B,C,D,*∀*x*∈Mor(*A*,*B*), ∀*y*∈Mor(*B*,*C*),∀*z*∈Mor(*C*,*D*), (*z*•*y*)•*x*=*z*•(*y*•*x*)

Again, an *automorphism* of an object *E*, is an isomorphism from *E* to itself.
Their set Aut(*E*) is the group of invertible elements of the monoid
End(*E*)=Mor(*E*,*E*).

- Hom(
*X*,*f*) = (Mor(*X*,*E*)∋*g*↦*f*•*g*), with target Mor(*X*,*F*) for any target*F*of*f*. - Hom
_{F}(*f*,*X*) = (Mor(*F*,*X*)∋*g*↦*g*•*f*), with target Mor(*E*,*X*). Simplified as Hom(*f*,*X*) in abstract categories where*f*determines*F*.

Hom(*X*, *g*) ০ Hom(*X*, *f*) =
Hom(*X*, *g*•*f*)

Hom_{F}(*f*, *X*) ০
Hom_{G}(*g*, *X*) =
Hom_{G}(*g*•*f*, *X*)

**Monomorphism**. In a category, a morphism
*f*∈Mor(*E*,*F*)
is called *monic*, or a *monomorphism*, if Hom(*X*,*f*)
is injective for all objects *X*:

∀*g*,*h*∈Mor(*X*,*E*),
*f*•*g* = *f*•*h* ⇒ *g* = *h*.

∀*g*,*h*∈Mor(*F*,*X*),
*g*•*f* = *h*•*f* ⇒ *g* = *h*.

In any concrete category, all injective morphisms are monic, and
any morphism with image *F* is *F*-epic.
However, the converses may not hold, and exceptions may be uneasy
to classify, especially as the condition depends on the whole
category.

**Sections, retractions.** When *g*•*f* = 1_{E}
we say that *f* is a section of *g*, and that *g* is a retraction of *f*.

- A morphism
*f*∈Mor(*E*,*F*) is a section (or*section in F*if the category is concrete), if 1_{E}∈Im(Hom_{F}(*f*,*E*)), i.e. ∃*g*∈Mor(*F*,*E*),*g*•*f*=1_{E}.

Then*f*is monic and for all objects*X*we have Im(Hom_{F}(*f*,*X*)) = Mor(*E*,*X*). -
A morphism
*g*∈Mor(*F*,*E*) is a retraction (or*retraction on E*if the category is concrete), if 1_{E}∈Im(Hom(*E*,*g*)), i.e. ∃*f*∈Mor(*E*,*F*),*g*•*f*=1_{E}.

Then*g*is epic and for all objects*X*we have Im(Hom(*X*,*g*)) = Mor(*X*,*F*).

- Hom
_{E}(*g*,*X*) is injective (*g*is epic) - ∀
*h*∈Mor(*E*,*X*),*h*=*h*•*g*•*f*= Hom_{F}(*f*,*X*)(*h*•*g*).

A morphism *f* is an isomorphism if and only if Hom(*X*,*f*) :
Mor(*X*,*E*) ↔ Mor(*X*,*F*); its inverse is then
Hom(*X*, *f*^{ -1}).

In concrete categories, Section ⇒ Injective morphism ⇒ Monomorphism

In categories of relational systems, Retraction ⇒ Quotient ⇒ Surjective morphism ⇒ Epimorphism

**Theorem.** Any small category is isomorphic to that of all morphisms in a family of typed algebras.

- The set
*T*of types copies the set of objects; one type per isomorphism class suffices. *L*= ∐_{t,t'∈T}Mor(*t'*,*t*) seeing Mor(*t'*,*t*) as a set of function symbols from*t*to*t'*.- Each object
*E*is interpreted as a typed set ∐_{t∈T}*t*where_{E}*t*= Mor(_{E}*t*,*E*).

Im

The existence of an isomorphism

∀

In particular for any monoid *M* there is a language
*L* of function symbols and an *L*-algebra *X*
such that End_{L} *X* is
isomorphic to *M*.

Any group is isomorphic to a permutation group, namely the group of automorphisms
of an algebra.

For any initial objectsBy this unique isomorphism,X,Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X),g•f∈ Mor(X,X) ∧ 1_{X}∈ Mor(X,X) ∴g•f= 1_{X}.

Similarly,f•g= 1_{Y}. Thusfis an isomorphism, unique because Mor(X,Y) is a singleton.∎

Similarly, an object

Such objects exist in many categories, but are not always interesting. For example, in any category of relational systems containing representatives (copies) of all possible ones with a given language:

- Singletons where relations are constantly true, are final objects in categories of one-type systems or algebras; for multi-type systems, final objects are made of one singleton per type.
- The empty set where Boolean constants are false is the initial object.

- Objects are all (
*X*,φ) where*X*is a set and φ:*X*×*K*→*B*; - Mor((
*X*,φ),(*Y*,φ') = {*f*∈*Y*| ∀^{X}*a*∈*X*,∀*k*∈*K*, φ(*a*,*k*) = φ'(*f*(*a*),*k*)}.

For any subset

(There exists a section with image *A*) ⇔ (for any object *N*, Mor(*A*,*N*) =
{*g*_{|A} | *g*∈Mor(*E*,*N*)})
⇒ (any embedding with image *A* is a section).

Let us generalize the concept of embedding to any concrete category *C*, making
it work the same as in categories of relational systems beyond the case of sections
(but unlike sections, it will require checking the whole category).
For any subset *A* of an object *E* of *C*, let To-*A* be the category where

- Objects are all
*C*-morphisms*f*into*A*, conceived as (*F*,*f*) where*F*is any object of*C*,*f*∈ Mor_{C}(*F*,*E*) and Im*f*⊂*A*; - Mor
_{To-A}((*F*,*f*),(*G*,*g*)) = {*h*∈Mor_{C}(*F*,*G*) |*f*=*g*০*h*}

Now a morphism *f* in *C* is an *embedding* if it is a final object of any
To-*A*. Then it is also a final
object of To-(Im *f*). It is an *embedding onto A* if moreover *A* = Im *f*.

All such embeddings, even non-injective ones, are monic : Section ⇒ Embedding ⇒ Monomorphism.

For any object *F*, Mor(*F*,*P*) ≅
∏_{i∈I} Mor(*F*,*E _{i}*)

Any data of

(∀*i*∈*I*, π_{i}∈Mor_{L}(*P*,*E _{i}*)) ⇔
(∀

In the more general case, assuming (Inc), the reverse inclusion not only defines Mor(

(*P*, φ) =
^{C}∏_{i∈I} *E _{i}*

For all *F*, ∏_{i∈I}
Hom (*F*,φ_{i}) :
Mor(*F*,*P*) ↔ ∏_{i∈I} Mor(*F*,*E _{i}*)

(*K*, *j*) = ^{C}∐_{i∈I}
*E _{i}*

For all *F*, ∏_{i∈I}
Hom_{K}(*j*_{i}, *F*) :
Mor(*K*,*F*) ↔ ∏_{i∈I} Mor(*E _{i}*,

- Objects are all (
*E*,*x*) where*X*is an object of*C*and*x*∈*E* - Mor((
*E*,*x*),(*F*,*y*)) = {*f*∈Mor(*E*,*F*) |*f*(*x*)=*y*}.

For any egg of an acting category, its image in the resulting concrete category is also an egg.

- Mor(
*X*,*Y*) ⊂ Mor_{M}(*X*,*Y*) - Mor(
*M*,*X*) = Mor_{M}(*M*,*X*)

The composition in

The last axiom of monoid,

∀

This monoid (

*h _{a}*০

**Proposition.** For a monoid (*M*,*e*, •) seen as an
*M*-set interpreting • as action, (*M*, *e*) is
an egg of the category of *M*-sets; other eggs are the (*X*,*x*)
where *x* is a free and generating element of *X*.

A

∀*u*∈*E ^{I}*, ∃!

This gives

Given an egg (

For any category

*L* = ∐_{n∈ℕ} *L _{n}*

Depending on *C*, a clone may exist or not,
but it is essentially unique, serving as the maximal algebraic language for objects of *C*.

- Each e
_{i,n}acts as the*i*-th projection from*E*^{n}to*E*, i.e. ∀*i*<*n*, ∀*u*∈*E*, e^{n}_{i,n}⋅*u*=*u*._{i} - ∀
*n*∈ℕ, ⃖φ_{E}^{(n)}:*E*→ Mor(^{n}*L*,_{n}*E*).

Mor(*E*,*F*) ⊂ Mor_{L}(*E*,*F*)

∀*n*∈ℕ, Mor(*L _{n}*,

- ∀
*x*∈*L*,_{n}*x*=*x*⋅1_{n}

3. implies that 1* _{n}* generates

- Constant symbols e
_{i,n} - An
*L*-algebra structure on each*L*, that is a sequence of operations we shall abusively denote the same • :_{n}*L*×_{p}*L*→_{n}^{p}*C*for each_{n}*p*∈ℕ.

∀*n*,*p*∈ℕ, ∀*s*∈*L _{p}*,
∀

A typed algebra

- 1. and 2. make
*L*an_{n}*L*-module. - As 3. was seen to imply ∀
*M*, ∀*f*∈Mor_{L}(*L*,_{n}*M*),*f*= ⃖φ_{M}^{(n)}(*f*০1), it completes making 1_{n}a basis of_{n}*L*, equivalently in the small category_{n}*L*with its*L*-morphisms or in its whole variety.

(*) I took the initiative to call it "egg" as I am not aware of any existing name for that concept in the literature.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

4. Model Theory