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

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

*L*⋆*E* = ∐_{s∈L}
*E ^{ns}*

Most often, we shall only use one

**E**={(*r*,*x*)∈*L*⋆*E* | *r*(*x*)}.

*r _{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**}.

For any relational language

It is the smallest

- A class of sets called
*objects*

- A class of functions called
*morphisms*; for any objects*E*,*F*we have a set Mor(*E*,*F*)⊂*F*^{E}of morphisms from*E*to*F*, that 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*०*f*∈Mor(*E*,*G*).

A relational symbol interpreted in a given concrete category is said to be

- the class of objects, regarded as pure elements (ignoring any inclusion order);
- the sets Mor(
*E*,*F*) between any two objects*E*,*F*, and regarded as pairwise disjoint; - the composition operations,
Mor(
*F*,*G*)×Mor(*E*,*F*)→Mor(*E*,*G*).

*r _{E}*
= {

However, even if we could take "all these structures" to turn these objects into relational systems (a possibility only ensured in small categories because of

Indeed, for any

- Substituting arguments of a
*r*∈*L*by a map σ to*n*' other variables (∀*E*,∀*x*∈*E*^{n'},*r'*(*x*)⇔*r*(*x*०σ)), works :*r'*(*x*) ⇒*r*(*x*०σ) ⇒*r*(*f*०*x*०σ) ⇒*r'*(*f*०*x*). - ∀
*r*,*r*'∈*L*,*n*=_{r}*n*⇒ ∀_{r'}*x*∈*E*^{nr}, (*r*(*x*)∧*r'*(*x*)) ⇒ (*r*(*f*०*x*)∧*r'*(*f*०*x*)) - ∀
*r*,*r*'∈*L*,*n*=_{r}*n*⇒∀_{r'}*x*∈*E*^{nr}, (*r*(*x*)∨*r'*(*x*)) ⇒ (*r*(*f*०*x*)∨*r'*(*f*०*x*)) - For 0 and 1 it is trivial

- ∀
*x*,*y*∈*E*,*x*=*y*⇒*f*(*x*)=*f*(*y*) - ∀
*x*∈*E*^{nr},(∃*y*∈*E*,*r*(*x*,*y*)) ⇒ (∃*z*=*f*(*y*)∈*F*,*r*(*f*०*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*०_{F}*f*=*h*where_{E}*h*:_{E}*E*→τ,*h*:_{F}*F*→τ are the functions giving the type of each object.

More texts on algebra

Back to homepage : 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. Special morphisms

3.3. Algebras

3.4. Algebraic terms and term algebras

3.5. Integers and recursion

3.6. Arithmetic with addition

4.1. Finiteness and countability

4.2. The Completeness Theorem

4.3. Infinity and the axiom of choice

4.4. Non-standard models of Arithmetic

4.5. How theories develop

4.6. The Incompleteness Theorem