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).
∃g∈Mor(F,E), g০f= IdE ∧
f০g= IdF, which will serve as definition in more
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).Automorphisms. An automorphism of an object E is an isomorphism of E to itself.
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.
∀(s,x,y)∈L'⋆E, f(y) = sF(f০x) = f(sE(x)) ⇒ y=sE(x).
fL-1 = (f-1)L ∴ φE০fL-1 = f -1০f০φE০fL-1 = f -1০φF০fL০fL-1 = f -1০φF.
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 ∈ MorL(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).Every isomorphism is an elementary embedding.
Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphismIf f∈ End(E) is an invariant elementary embedding then it is an automorphism:
Im f is also invariant (defined by ∃y∈ E, f(y)=x)Elementary equivalence. Different systems are said to be elementarily equivalent, if they have all the same true ground first-order formulas. The existence of an elementary embedding between systems implies that they are elementarily equivalent.
∀x∈ E, x∈Im f ⇔ f(x)∈Im f
Im f = 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.
3.1. Morphisms of relational systems and concrete categories4. Model Theory
3.3. Special morphisms
3.5. Actions of monoids
3.7. Algebraic terms and term algebras
3.8. Integers and recursion
3.9. Arithmetic with addition