For simplicity, let us focus the study on systems with only one type.

For any numberThe sets of all

- Rel
_{E }^{(n)}= ℘(*E*)^{n} - Op
_{E}^{(n)}=*E*^{En}

- A
*relational language*if its symbols aim to represent relations - An
*algebraic language*if its symbols aim to represent operations.

A relational language

∏_{s∈L} ℘(*E ^{ns}*)
≅ ℘(

Most often, we shall only use one

**E**={(*s*,*x*)∈*L*⋆*E* | *s*(*x*)}.

*s _{E}* =
{

Mor_{L}(*E*,*F*) = {*f*
∈*F ^{E}*|∀

= {

Mor_{L}(*E*,*F*) = {*f*
∈*F*^{E}| *f*_{L}[**E**]⊂**F**} =
{*f* ∈*F*^{E}| **E**⊂*f*_{L}***F**}.

- A class of sets called
*objects*

- A class of functions called
*morphisms*. For any objects*E*,*F*, the set Mor(*E*,*F*)⊂*F*^{E}of all morphisms from*E*to*F*, is the set of all functions from*E*to*F*which are morphisms.

- 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 interpreted in a given concrete category is said to be

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

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

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*
running over *s* (with variable *K*).

This can be easily deduced from the fact that any union of preserved structures in a
concrete category is a preserved structure (not only finite unions but unions of families
indexed by any set). Any intersection of a family of preserved structures
is also a preserved structure.

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*))

However morphisms may no more preserve structures defined with other symbols (¬,⇒,∀).

- 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*anf*F* - A function
*f*:*E*→*F*that is a morphism when regarding τ as a list of unary relation symbols (by the same idea as the use of classes instead of types in set theory); or equivalently, 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 object.

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1.4. Model TheoryMorphisms of relational systems and concrete categories

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