5.2. First-order invariants in concrete categories
Multiple automorphisms and models
Most geometries, except RCF, have a big group
of automorphisms in each model.
To produce such spaces from a framework without automorphisms
(ZF with only sets and no pure elements, or second-order arithmetic),
requires not only developments
but also to forget some structures, and notice how some permutations then become
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 ↦ f০g 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.
The Galois connection (Aut, sInv)
For any set E, the relation of strong
preservation between its set RelE 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) ⊂ RelE
For any subgroup G of ⤹E (automorphism group in any category),
sInv G = Inv G.
∀L⊂RelE, ∀P⊂ ⤹E, L ⊂ sInv
P ⇔ P ⊂ AutL E.
Any relation defined
by an expression with language L⊂R without parameters
is invariant by automorphisms, so is in Inv(AutL E).
This means that adding it to L preserves AutL E
(like other sets of isomorphisms between models).
Constructions leave unchanged the automorphism groups seen as abstract
groups, providing more types on which they act.
The double meaning of "invariance"
Unlike provability which by the completeness
theorem ranges over all truths of models, definability by language L without parameters
may not fill Inv(AutL(E)), for the following reason.
Structures r ∈ Inv(AutL(E)), distinguishing any
tuple t∈r from those outside r, are those for which no element of
AutL(E) can move t out. This suggests
that t is not similar to anyone outside r with respect to L, unless
the trouble is to choose
an automorphism witnessing the similarity.
But this dissimilarity, and thus r itself, may be formally inexpressible from L
as it may require infinitely complex descriptions.
For example, there is no nontrivial automorphism in ℝ, or in a model of second-order arithmetic,
yet not all objects of an uncountable
type (real numbers or sets of integers) can be defined without parameter
(but, as objects, they are definable with a parameter: themselves).
Both concepts of invariance can be still reconciled in different ways:
A set of axioms of a theory can be seen as a mere tool to approach an intended
range of models, on the other side of the
Similarly, a language may be a mere tool to approach an intended concept
of invariance better defined by a group G on the other side of the (Aut, sInv)
connection. Specifying the geometry of a space E by a permutation group
G of E intended as Aut(E), was a core idea of Felix Klein's
Thus, "invariance" in geometry will be
meant by G, and equivalently qualify structures defined with
parameters in ℝ or among structures over ℝ only.
- 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
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.
Basis and algebraic structures
We already gave the construction
of all invariant relations, elements of Inv(End E) or Inv(Aut E), from the
trajectories or orbits of tuples.
Algebraic structures come as functional cases of relations defined by tuples
(like with categories of acts
and interpretations of drafts in
A basis of a space X is a subset B⊂X such that
(X,IdB) is an initial object of the category of
all (E,u) where E is a space and u ∈EB:
This gives X the role of the set of all possible B-ary operation symbols
interpreted in each E and preserved in the given category.
Its essential uniqueness means that any equinumerous basis
in the same concrete category, give their spaces the role of the same set of
Given a space M with a one-element basis e, a space X with
basis B plays the role of the B-ary coproduct (of the constant family),
M, with the ji ∈ Mor(M,X) defined for each
i∈B by ji(e) = i.
These ji are sections.
Conversely from any chosen object M in a category C, if (X, B)
with B ∈ Mor(M,X)I
is a coproduct ∐i∈I M then B is a
basis of MX in the concrete category
The clone of a concrete category C (usually called an
abstract clone) is an algebraic language
L = ∐n∈ℕ Cn, where each set
Cn of n-ary symbols is an object with a chosen basis
of n elements 1n =
∈ Cnn in C.
It is essentially unique for a given C, serving as the maximal algebraic
language for objects of C. This extends to all arities the concept of monoid
seen as a language of function
Then, each Cn is the n-ary
C1 in C.
Each object E of C, gets the L-algebra structure
φE = ∐n∈ℕ
φE(n), defined taking
as ⃖φE(n) the inverse of
Mor(Cn,E) ∋ f ↦
f০1n which is bijective to En because
1n is a basis in C. This is expressible by both axioms
(extending those for categories of acts)
This implies (for all objects E, F),
- Each ei,n acts as the i-th
projection from En to E, i.e. ∀i<n,
ei,n ⋅ u = ui.
- ∀n∈ℕ, ⃖φE(n) :
En → Mor(Cn,E).
Mor(E,F) ⊂ MorL(E,F)
Let us recall the proof of MorL(Cn,E)
⊂ Mor(Cn,E) previously seen for n=1. From
1n being a basis of Cn and
IdCn ∈ Mor(Cn,Cn), comes
∀n∈ℕ, Mor(Cn,E) =
(∀x∈Cn, f(x) = f(x⋅1n) = x⋅
(f০1n) ∴ f =
- ∀x∈Cn, x = x⋅1n
3. implies that 1n generates Cn as
〈1n〉Cn = Cn.
VarietiesOne may conceive an abstract clone L without any a priori data of a category C.
First we may take as C the small category made of the sequence of objects
(Cn)n∈ℕ. The structure of L, defined by the
morphisms between these objects only, can be presented as that of a typed algebra,
with types Cn and operation symbols:
Between other objects, thus L-algebras, one may choose the sets of morphisms as the widest
possibility Mor(E,F) = MorL(E,F).
- the constant symbols ei,n
- an L-algebra structure on each Cn,
formalizable as a sequence of operations ⋅ : Cp
× Cnp → Cn
for each p∈ℕ.
An L-module (or representation of L),
is an L-algebra M satisfying 1. and 2. where in 2., Mor is defined
as MorL :
(s⋅x) ⋅ u = s ⋅ ((xi
The category of all L-modules with Mor = MorL, that is the maximal
choice of C for a given clone, is called a variety. It extends
to all arities the concept of action.
We shall study later the
general concept of variety, defined as the category
of all L-algebras satisfying conditions eventually expressible like this one of being
modules, but without assuming L to be an abstract clone: L may be formed of
domains with basis which need not be one per arity; they need not belong to this category
(but are seen as additional objects to this category for making sense of the basis), and even
need not be algebras. Instead, with a simple algebraic language L, conditions may be
expressed by a list of axioms made of ∀ over equalities of L-terms.
As will be shown, such categories are still qualified as varieties in the sense of
having an abstract clone ∐n∈ℕ Cn, and being
identifiable with the variety of this clone; L may differ from it, but generates it in
the sense that each L-algebra Cn is L-generated by
(the image of) its basis, i.e. is the set of all symbols definable by n-ary L-terms
(where the role of the symbols of variable is played by the symbols
∀n∈ℕ, 〈1n〉L = Cn.
For a typed algebra L = ∐n∈ℕ Cn with the
above mentioned symbols (thus made of L-algebras Cn), it is qualified
as an abstract clone if each Cn satisfies the above axioms 1., 2. and 3. :
These axioms generalize the 3 axioms of monoid
(as C1 is a monoid).
- 1. and 2. make Cn an L-module.
- 3. makes 1n a basis of Cn
in this variety, by ∀M, ∀f∈MorL(Cn,M),
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(AutL(E))
by means of extended kinds
of "definitions" (with infinite amounts of data) has been
investigated by Borner, Martin Goldstern, and Saharon Shelah : short
presentation in .ps - full article.
Set Theory and Foundations of Mathematics