**Strong preservation**. A function *f* ∈ *F ^{E}* is said to

**Isomorphism**. An isomophism between algebras is a bijective embedding.
Generally, an *isomorphism* between objects *E* and *F* of a concrete category,
is a morphism (*f* ∈Mor(*E*,*F*)) that is bijective (*f*
: *E* ↔ *F*) and whose inverse is a morphism (*f*^{ -1}∈Mor(*F*,*E*)).

Two objects *E*, *F* are said to be *isomorphic*
if there exists an isomorphism between them. This is an equivalence predicate, i.e. it works as an
equivalence relation on the universe.

The *isomorphism class* of an object in a category, is the class of all objects (in this category) 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
any object in it (independently of the choice).

Injective embeddings are isomorphisms to their images.

∀(*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}*

More generally for relational systems, 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.

Thus, they also preserve invariant structures defined using symbols of

Every isomorphism is an elementary embedding.

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 *E*, ∀*f*∈*E ^{E}*, ∀

So we have: Automorphism ⇔ (Endomorphism ∧ Isomorphism)

However an endomorphism of *E* which is an isomorphism to a strict
subset of *E*, is not an automorphism.

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

∀x∈E,x∈Imf⇔f(x)∈Imf

Imf=E. ∎

It is the smallest

Set theory and foundations of mathematics

1. First foundations of mathematics

2. Set theory (continued)

3.1. Morphisms of 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 and term algebras

3.9. Integers and recursion

3.10. Arithmetic with addition