We shall call

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*⋆(*E*/*R*) so that the
compatibility condition means that the quotient is also an algebra.

For any *L*'-systems (*E*,**E**) and (*F*,**F**), any
*f* : *E*→*F*, *B*⊂*F* and *A*= *f**(*B*) we have

(*f*∈Mor(*E*,*F*) ∧ *B*∈Sub_{L}*F*) ⇒ *A*∈Sub_{L}*E*

(**F**⊂*f*_{L}[**E**] ∧ *A*∈Sub_{L}*E*) ⇒ *B*∈Sub_{L}*F*

*f*_{L}[**E**]=**F** ⇒ (*B*∈Sub_{L}*F* ⇔ *A*∈Sub_{L}*E*)

∀*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).

∀(*s*,*x*,*y*)∈*L'*⋆*E*, *f*(*y*)
= *s _{F}*(

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

*f _{L}*

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.

For any set

∀*G*⊂*E ^{E}*, ∀

Automorphism ⇔ (Endomorphism ∧ Isomorphism)

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

∀x∈E,x∈Imf⇔f(x)∈ Imf

Imf=E. ∎

Set theory and foundations of mathematics

1.First foundations of mathematics

2. Set theory (continued)

3.1. 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