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∈SubLF) ⇒ A∈SubLE
(F⊂fL[E] ∧ A∈SubLE) ⇒ B∈SubLF
fL[E]=F ⇒ (B∈SubLF ⇔ A∈SubLE)
∀x∈Enr, x∈rE ⇔ f০x∈rF.Embeddings. An f ∈MorL(E,F) is called an L-embedding if it strongly preserves all structures : E = fL*(F).
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) = sF(f০x) = f(sE(x)) ⇒ y=sE(x).
fL-1 = (f -1)L ∴ φE০fL-1 = f -1০f০φE০fL-1 = f -1০φF০fL০fL-1 = f -1০φF.
Isomorphism ⇒ Elementary embedding ⇒ Embedding ⇒ Injective morphismElementary 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.
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.
∀G⊂EE, ∀F⊂℘(E), G ⊂ EndFE ⇔ (∀f∈G, ∀A∈F, f[A] ⊂ A) ⇔ F ⊂ SubGE.An automorphism of an object E is an isomorphism of E to itself:
Automorphism ⇔ (Endomorphism ∧ Isomorphism)An endomorphism f∈ End(E) may be an embedding but not an automorphism : just an isomorphism to a strict subset of E. But any endomorphism which is an invariant elementary embedding is an automorphism:
Im f is also invariant (defined by ∃y∈ E, f(y)=x)
∀x∈ E, x∈Im f ⇔ f(x)∈ Im f
Im f = E. ∎
3.1. Relational systems and concrete categories4. Model Theory
3.3. Special morphisms
3.5. Actions of monoids
3.6. Invertibility and groups
3.8. Algebraic terms and term algebras
3.9. Integers and recursion
3.10. Arithmetic with addition