**Isomorphism**.
In any concrete category, an *isomorphism* between objects *E* and *F*, is a morphism *f* ∈Mor(*E*,*F*)
such that *f*
: *E* ↔ *F* and *f*^{ -1}∈Mor(*F*,*E*).
Or equivalently,
∃*g*∈Mor(*F*,*E*), *g*০*f*= Id_{E} ∧
*f*০*g*= Id_{F}, which will serve as definition in more
general categories.

Two objects *E*, *F* are said to be *isomorphic*
if there exists an isomorphism between them.

**Endomorphisms.** An *endomorphism* of an object *E*
in a category, is a morphism from *E* to itself. Their set
is written End(*E*)=Mor(*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.

Every injective morphism

∀(*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.

Injective embeddings are isomorphisms to their images.

Embeddings still strongly preserve structures defined using the
symbols in *L* and the logical symbols ∧,∨,0,1,¬, and also = in
the case of injective embeddings.

Thus, they also preserve invariant structures defined
using symbols of *L* and ∧,∨,¬,0,1,∃ where any
occurrence of ¬ comes after (inside) any occurrence of ∃.

**Elementary embedding.** An *f* ∈
Mor_{L}(*E*,*F*) is called an *elementary embedding*
(or *elementary* *L*-*embedding*) if it (strongly) preserves all
invariant structures (defined
by first-order formulas with language *L*).

Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphism

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

∀x∈E,x∈Imf⇔f(x)∈Imf

Imf=E. ∎

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.

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. Categories

3.7. Algebraic terms and term algebras

3.8. Integers and recursion

3.9. Arithmetic with addition