Spaces with non-trivial automorphisms will be interestingly seen as
non-unique in each isomorphic class (instead of picking one to represent them all),
for the following reason. Any isomorphism
*f* between 2 systems *E* and *F* induces a bijection
*g* ↦ *f*০*g* from Aut(*E*) to the set of isomorphisms
between *E* and *F*. Thus, a plurality of automorphisms
of *E* means a plurality of isomorphisms between *E*
and *F*. This plurality makes the difference between *E* and *F*
meaningful, as an object in *E* may correspond to several possible ones in
*F*, depending on the choice of isomorphism, so choosing an object in *E* does
not mean choosing an object in *F* in any invariant way. By contrast, the uniqueness of
isomorphism between standard models of ℕ or ℝ makes them unambiguously
play the role of each other (copies of each other), making their plurality superfluous.

For any set *E*, the relation of strong
preservation between its set Rel_{E} of relations and its set ⤹*E*
of permutations, defines a Galois connection
(Aut, sInv) similar to (End, Inv), where sInv
gives the set of *strong invariants* :

∀*P*⊂⤹*E*, sInv *P* = Inv (*P*
∪ -*P*) = Inv (*P*) ∩ Inv(-*P*) ⊂ Rel_{E}

∀*L*⊂Rel_{E}, ∀*P*⊂ ⤹*E*, *L* ⊂ sInv
*P* ⇔ *P* ⊂ Aut_{L} *E*.

Any relation defined by an expression with language

Constructions leave unchanged the automorphism groups seen as abstract groups, providing more types on which they act.

Structures

Both concepts of invariance can be still reconciled in different ways:

- For first-order logic, non-trivial automorphisms may externally exist over non-standard models of second-order arithmetic, thus also non-standard real closed fields (moving undefinable real numbers to infinitely close neighbors);
- In second-order logic, we may admit "second-order" ways of defining structures where any subset of ℕ is "defined" by the infinite data of all its elements(*).

Now starting with a mere concrete category of intended spaces whose class of morphisms is given by intuition, let us review methods to produce invariant structures.

Algebraic structures come as functional cases of relations defined by tuples (like with categories of acts and interpretations of drafts in algebras):

A

∀*u*∈*E ^{B}*, ∃!

Given a space

Conversely from any chosen object

Then, each *C _{n}* is the

- Each e
_{i,n}acts as the*i*-th projection from*E*^{n}to*E*, i.e. ∀*i*<*n*, ∀*u*∈*E*, e^{n}_{i,n}⋅*u*=*u*._{i} - ∀
*n*∈ℕ, ⃖φ_{E}^{(n)}:*E*→ Mor(^{n}*C*,_{n}*E*).

Mor(*E*,*F*) ⊂ Mor_{L}(*E*,*F*)

∀*n*∈ℕ, Mor(*C _{n}*,

- ∀
*x*∈*C*,_{n}*x*=*x*⋅1_{n}

3. implies that 1* _{n}* generates

- the constant symbols e
_{i,n} - an
*L*-algebra structure on each*C*, formalizable as a sequence of operations ⋅ :_{n}*C*×_{p}*C*→_{n}^{p}*C*for each_{n}*p*∈ℕ.

An

∀*n*,*p*∈ℕ, ∀*s*∈*C _{p}*,
∀

We shall study later the general concept of variety, defined as the category of all

∀*n*∈ℕ, 〈1* _{n}*〉

- 1. and 2. make
*C*an_{n}*L*-module. - 3. makes 1
a basis of_{n}*C*in this variety, by ∀_{n}*M*, ∀*f*∈Mor_{L}(*C*,_{n}*M*),*f*= ⃖φ_{M}^{(n)}(*f*০1)._{n}

Equivalently, an algebraic language is an abstract clone when, for some given class of interpretations (algebras), it is stable by definitions by terms, in the sense that any two symbols keeping the same interpretation throughout this class are confused.

We already saw two examples of varieties

- That of all
*L*-algebras for any given algebraic language*L*; its clone is the sequence of term algebras with each finite arity. - That of all monoids; its clone is made of free monoids.

(*)This bridge to Inv(Aut

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