The profinite theory of rational languages
Laure Daviaud
LIP, ENS Lyon
Toulouse, 22th June 2016
The 3 reasons I am here...
2/20
The 3 reasons I am here...
1 - Topology: metric space, limits of sequences of words...
2/20
The 3 reasons I am here...
1 - Topology: metric space, limits of sequences of words...
2 - Languages: classes of rational languages...
2/20
The 3 reasons I am here...
1 - Topology: metric space, limits of sequences of words...
2 - Languages: classes of rational languages...
3 - Toulouse...
2/20
Profinite theory
→ a topological approach for the study of rational languages.
3/20
Profinite theory
→ a topological approach for the study of rational languages.
A: finite alphabet
3/20
Profinite theory
→ a topological approach for the study of rational languages.
A: finite alphabet
⊆
A∗ : set of words
L: rational language
3/20
Profinite theory
→ a topological approach for the study of rational languages.
A: finite alphabet
⊆
(A∗ , d): metric space
⊆
A∗ : set of words
L: rational language
L: subset satisfying some
topological properties
3/20
Profinite theory
→ a topological approach for the study of rational languages.
A: finite alphabet
⊆
(A∗ , d): metric space
⊆
A∗ : set of words
L: rational language
L: subset satisfying some
topological properties
automata
logic
rational expressions
monoids
3/20
Profinite theory
→ a topological approach for the study of rational languages.
A: finite alphabet
⊆
(A∗ , d): metric space
⊆
A∗ : set of words
L: rational language
L: subset satisfying some
topological properties
automata
logic
rational expressions
monoids
3/20
Finite monoids and rational languages
A few things to know about monoids...
Monoid: a set with an associative operation and a neutral element.
Idempotent: e 2 = e
bla In a finite monoid, every element has a unique idempotent power
bla x ∈ M −→ x |M|! is idempotent
4/20
Finite monoids and rational languages
A few things to know about monoids...
Monoid: a set with an associative operation and a neutral element.
Idempotent: e 2 = e
bla In a finite monoid, every element has a unique idempotent power
bla x ∈ M −→ x |M|! is idempotent
A monoid M recognises a language L if there is a morphism
ϕ : A∗ → M and P ⊆ M s.t. L = ϕ−1 (P).
M
⊆
⊆
ϕ
A∗
ϕ−1 (P) = L
P
4/20
Finite monoids and rational languages
A few things to know about monoids...
Monoid: a set with an associative operation and a neutral element.
Idempotent: e 2 = e
bla In a finite monoid, every element has a unique idempotent power
bla x ∈ M −→ x |M|! is idempotent
A monoid M recognises a language L if there is a morphism
ϕ : A∗ → M and P ⊆ M s.t. L = ϕ−1 (P).
M
⊆
⊆
ϕ
A∗
ϕ−1 (P) = L
P
A language is rational iff it is recognised by a finite monoid.
4/20
Examples and syntactic monoid
M
⊆
⊆
ϕ
A∗
ϕ−1 (P) = L
P
Example: L = {w ∈ A∗ | |w | is even}
5/20
Examples and syntactic monoid
M
⊆
⊆
ϕ
A∗
ϕ−1 (P) = L
P
Example: L = {w ∈ A∗ | |w | is even}
(Z/2Z, +) - ϕ(A) = {1}
ϕ
ϕ−1 ({0}) = L
Z/2Z
⊆
⊆
A∗
{0}
5/20
Examples and syntactic monoid
M
⊆
⊆
ϕ
A∗
ϕ−1 (P) = L
P
Example: L = {w ∈ A∗ | |w | is even}
(Z/2Z, +) - ϕ(A) = {1}
ϕ
ϕ−1 ({0}) = L
Z/2Z
⊆
⊆
A∗
{0}
Syntactic monoid: the smallest monoid recognising L.
= Monoid of transitions of a minimal deterministic automaton.
5/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
Syntatic monoid of {u} (or {v }...)
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
Syntatic monoid of {u} (or {v }...)
Example 2: a ∈ A - separate a99 and a100 ?
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
Syntatic monoid of {u} (or {v }...)
Example 2: a ∈ A - separate a99 and a100 ?
Z/2Z
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
Syntatic monoid of {u} (or {v }...)
Example 2: a ∈ A - separate a99 and a100 ?
Z/2Z
Example 3: u ∈ A∗ , n ∈ N - separate u n! and u (n+1)! ?
6/20
Separation of words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
Example 1: separate u 6= v ?
Syntatic monoid of {u} (or {v }...)
Example 2: a ∈ A - separate a99 and a100 ?
Z/2Z
Example 3: u ∈ A∗ , n ∈ N - separate u n! and u (n+1)! ?
x ∈ M then x |M|! = x (|M|+1)! = the idempotent power of x in M
=⇒ ϕ(u)|M|! = ϕ(u)(|M|+1)!
u n! and u (n+1)! cannot be separated
by a monoid of size less than n
6/20
Distance over words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
7/20
Distance over words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
d(u, v ) = 2−n
where n is the minimal size of a monoid that separates u and v .
7/20
Distance over words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
d(u, v ) = 2−n
where n is the minimal size of a monoid that separates u and v .
d is an ultrametric distance:
d(u, v ) = 0 iff u = v
d(u, v ) = d(v , u)
d(u, v ) 6 max(d(u, w ), d(w , v ))
7/20
Distance over words
A monoid M separates u and v if:
there is a morphism ϕ : A∗ → M such that ϕ(u) 6= ϕ(v ).
d(u, v ) = 2−n
where n is the minimal size of a monoid that separates u and v .
d is an ultrametric distance:
d(u, v ) = 0 iff u = v
d(u, v ) = d(v , u)
d(u, v ) 6 max(d(u, w ), d(w , v ))
The words u n! and u (n+1)! are closer and closer...
7/20
Profinite monoid
Definition
c∗ :
Profinite monoid A
∗
completion of A with respect to the distance d.
Monoid if u and v sequences of words, (u.v )n = un vn
Metric space
A∗ dense subset
Compact
8/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
9/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
Zero (Reilly-Zhang 2000, Almeida-Volkov 2003)
|A| > 2
u0 , u1 , . . . an enumeration of the words of A∗
v0 = u0 , vn+1 = (vn un+1 vn )(n+1)!
ρA = n→∞
lim vn
9/20
Profinite monoid and rational languages
Universal property
M a finite monoid.
Every morphism ϕ : A∗ → M can be uniquely extended to a
c∗ → M.
continuous morphism ϕb : A
M
⊆
b
ϕ
⊆
c∗
A
b −1 (P) = L
ϕ
P
10/20
Profinite monoid and rational languages
Universal property
M a finite monoid.
Every morphism ϕ : A∗ → M can be uniquely extended to a
c∗ → M.
continuous morphism ϕb : A
M
⊆
b
ϕ
⊆
c∗
A
b −1 (P) = L
ϕ
P
10/20
Profinite monoid and rational languages
Universal property
M a finite monoid.
Every morphism ϕ : A∗ → M can be uniquely extended to a
c∗ → M.
continuous morphism ϕb : A
M
⊆
b
ϕ
⊆
c∗
A
b −1 (P) = L
ϕ
P
10/20
Profinite monoid and rational languages
Universal property
M a finite monoid.
Every morphism ϕ : A∗ → M can be uniquely extended to a
c∗ → M.
continuous morphism ϕb : A
M
⊆
b
ϕ
⊆
c∗
A
b −1 (P) = L
ϕ
P
c∗ .
A language L is rational iff L is open and closed in A
10/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
11/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
−→ For all morphisms ϕ : A∗ → M (finite monoid):
b ω ) is the idempotent power of ϕ(u)
b
−→ ϕ(u
in M.
11/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
−→ For all morphisms ϕ : A∗ → M (finite monoid):
b ω ) is the idempotent power of ϕ(u)
b
−→ ϕ(u
in M.
Zero (Reilly-Zhang 2000, Almeida-Volkov 2003)
|A| > 2
u0 , u1 , . . . an enumeration of the words of A∗
v0 = u0 , vn+1 = (vn un+1 vn )(n+1)!
ρA = n→∞
lim vn
11/20
V.I.P. words (very important profinite words)
Idempotent power
u ω = n→∞
lim u n!
−→ For all morphisms ϕ : A∗ → M (finite monoid):
b ω ) is the idempotent power of ϕ(u)
b
−→ ϕ(u
in M.
Zero (Reilly-Zhang 2000, Almeida-Volkov 2003)
|A| > 2
u0 , u1 , . . . an enumeration of the words of A∗
v0 = u0 , vn+1 = (vn un+1 vn )(n+1)!
ρA = n→∞
lim vn
−→ For all morphisms ϕ : A∗ → M (finite monoid):
b A ) = 0.
−→ if M has a zero then ϕ(ρ
11/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
12/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
A monoid T satisfies a word-identity u = v with u, v ∈ A∗ , if for all
morphisms ϕ : A∗ → T , ϕ(u) = ϕ(v ).
12/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
A monoid T satisfies a word-identity u = v with u, v ∈ A∗ , if for all
morphisms ϕ : A∗ → T , ϕ(u) = ϕ(v ).
Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff]
12/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
A monoid T satisfies a word-identity u = v with u, v ∈ A∗ , if for all
morphisms ϕ : A∗ → T , ϕ(u) = ϕ(v ).
Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff]
Pseudovariety of finite monoids: class of finite monoids closed under:
finite direct product
submonoid
quotient
12/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
A monoid T satisfies a word-identity u = v with u, v ∈ A∗ , if for all
morphisms ϕ : A∗ → T , ϕ(u) = ϕ(v ).
Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff]
Pseudovariety of finite monoids: class of finite monoids closed under:
finite direct product
submonoid
quotient
Varieties of finite monoids ←→ varieties of rational languages [Eilenberg]
12/20
Study of classes of rational languages
Birkhoff variety of monoids: class of monoids closed under:
direct product
submonoid
ϕ
quotient N quotient of M: M −
→ N with ϕ a surjective morphism.
A monoid T satisfies a word-identity u = v with u, v ∈ A∗ , if for all
morphisms ϕ : A∗ → T , ϕ(u) = ϕ(v ).
Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff]
Pseudovariety of finite monoids: class of finite monoids closed under:
finite direct product
submonoid
quotient
Varieties of finite monoids ←→ varieties of rational languages [Eilenberg]
→ Equations for pseudovarieties? Profinite equations! [Reiterman]
12/20
Classes of rational languages
Lattice (union, intersection)
Boolean algebra (lattice, complement)
Lattice closed under quotient
Boolean algebra closed under quotient
quotient : u −1 Lv −1 = {w | uwv ∈ L}
13/20
Equations
Definition
Given two profinite words u, v , a rational language L satisfies
u→v
if u ∈ L̄ implies v ∈ L̄
a, b ∈ A
Equation ab → aba
{L ⊆ A∗ | ab ∈
/ L} ∪ {L ⊆ A∗ | ab, aba ∈ L}
14/20
Equations
Definition
Given two profinite words u, v , a rational language L satisfies
u↔v
if u ∈ L̄ if and only if v ∈ L̄
a, b ∈ A
Equation ab ↔ aba
{L ⊆ A∗ | ab, aba ∈
/ L} ∪ {L ⊆ A∗ | ab, aba ∈ L}
14/20
Equations
Definition
Given two profinite words u, v , a rational language L satisfies
u6v
if for all w , w 0 ∈ A∗ , wuw 0 ∈ L̄ implies wvw 0 ∈ L̄
a, b ∈ A
Equation ab 6 aba
{L ⊆ A∗ | for all w , w 0 , if wabw 0 ∈ L then wabaw 0 ∈ L}
14/20
Equations
Definition
Given two profinite words u, v , a rational language L satisfies
u=v
if for all w , w 0 ∈ A∗ , wuw 0 ∈ L̄ if and only if wvw 0 ∈ L̄
a, b ∈ A
Equation ab = aba
{L ⊆ A∗ | for all w , w 0 , wabw 0 ∈ L iff wabaw 0 ∈ L}
14/20
Characterisation by equations on profinite words
Theorem [Gehrke, Grigorieff, Pin 2008]
Classes of rational languages
Lattice (union, intersection): →
Boolean algebra (lattice, complement): ↔
Lattice closed under quotient: 6
Boolean algebra closed under quotient: =
quotient : u −1 Lv −1 = {w | uwv ∈ L}
15/20
Examples
A alphabet
Example 1: Commutative languages
A language L is commutative if for all u ∈ L, com(u) ⊆ L.
16/20
Examples
A alphabet
Example 1: Commutative languages
A language L is commutative if for all u ∈ L, com(u) ⊆ L.
uv = vu
→ Boolean algebra closed under quotient
Decidability?
16/20
Examples
A alphabet
Example 1: Commutative languages
A language L is commutative if for all u ∈ L, com(u) ⊆ L.
uv = vu
→ Boolean algebra closed under quotient
Decidability?
A language L satisfies u = v if and only if ϕ(u) = ϕ(v ) in M where
M is the syntactic monoid of L and ϕ is its syntactic morphism
16/20
Examples
A alphabet
Example 1: Commutative languages
A language L is commutative if for all u ∈ L, com(u) ⊆ L.
uv = vu
→ Boolean algebra closed under quotient
Decidability?
A language L satisfies u = v if and only if ϕ(u) = ϕ(v ) in M where
M is the syntactic monoid of L and ϕ is its syntactic morphism
Example 2: Existence of a zero
16/20
Examples
A alphabet
Example 1: Commutative languages
A language L is commutative if for all u ∈ L, com(u) ⊆ L.
uv = vu
→ Boolean algebra closed under quotient
Decidability?
A language L satisfies u = v if and only if ϕ(u) = ϕ(v ) in M where
M is the syntactic monoid of L and ϕ is its syntactic morphism
Example 2: Existence of a zero
{ρA u = uρA = ρA | u ∈ A∗ }
16/20
Generalised star-height problem
Rational expressions:
1, a ∈ A,
E ∪ F , E ∩ F , E .F , c E , E ∗
17/20
Generalised star-height problem
Rational expressions:
1, a ∈ A,
E ∪ F , E ∩ F , E .F , c E , E ∗
Given a rational language L, what is the minimal number of nested
stars needed to describe L by such an expression?
17/20
Generalised star-height problem
Rational expressions:
1, a ∈ A,
E ∪ F , E ∩ F , E .F , c E , E ∗
Given a rational language L, what is the minimal number of nested
stars needed to describe L by such an expression?
→ Star-height 0 [Schützenberger, McNaughton-Papert]
Star-free languages, aperiodic monoid x ω+1 = x ω , FO[<]
17/20
Generalised star-height problem
Rational expressions:
1, a ∈ A,
E ∪ F , E ∩ F , E .F , c E , E ∗
Given a rational language L, what is the minimal number of nested
stars needed to describe L by such an expression?
→ Star-height 0 [Schützenberger, McNaughton-Papert]
Star-free languages, aperiodic monoid x ω+1 = x ω , FO[<]
→ Star-height 1
Example: (aa)∗ - Is there a nontrivial identity for this class ?
17/20
Equations for u ∗
Pu =
(joint work with Charles Paperman)
[
u∗p
and
Su =
p prefix of u
[
su ∗
s suffix of u
x ω y ω = 0 for x , y ∈ A∗ such that xy 6= yx
(E1 )
∗
x y = 0 for x , y ∈ A such that y ∈
/ Px
(E2 )
yx ω = 0 for x , y ∈ A∗ such that y ∈
/ Sx
(E3 )
x ω 6 1 for x ∈ A∗
(E4 )
x ` ↔ x ω+` for x ∈ A∗ , ` > 0
(E5 )
x → x ` for x ∈ A∗ , ` > 0
(E6 )
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
ω
DECIDABLE
18/20
Equations for u ∗
Pu =
(joint work with Charles Paperman)
[
u∗p
and
Su =
p prefix of u
[
su ∗
s suffix of u
x ω y ω = 0 for x , y ∈ A∗ such that xy 6= yx
(E1 )
∗
x y = 0 for x , y ∈ A such that y ∈
/ Px
(E2 )
yx ω = 0 for x , y ∈ A∗ such that y ∈
/ Sx
(E3 )
x ω 6 1 for x ∈ A∗
(E4 )
x ` ↔ x ω+` for x ∈ A∗ , ` > 0
(E5 )
x → x ` for x ∈ A∗ , ` > 0
(E6 )
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
ω
DECIDABLE Lattice
18/20
Equations for u ∗
Pu =
(joint work with Charles Paperman)
[
u∗p
and
[
Su =
p prefix of u
su ∗
s suffix of u
x ω y ω = 0 for x , y ∈ A∗ such that xy 6= yx
(E1 )
∗
x y = 0 for x , y ∈ A such that y ∈
/ Px
(E2 )
yx ω = 0 for x , y ∈ A∗ such that y ∈
/ Sx
(E3 )
x ω 6 1 for x ∈ A∗
(E4 )
x ` ↔ x ω+` for x ∈ A∗ , ` > 0
(E5 )
x → x ` for x ∈ A∗ , ` > 0
(E6 )
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
ω
DECIDABLE Lattice closed under quotients
18/20
Equations for u ∗
Pu =
(joint work with Charles Paperman)
[
u∗p
and
Su =
p prefix of u
[
su ∗
s suffix of u
x ω y ω = 0 for x , y ∈ A∗ such that xy 6= yx
(E1 )
∗
x y = 0 for x , y ∈ A such that y ∈
/ Px
(E2 )
yx ω = 0 for x , y ∈ A∗ such that y ∈
/ Sx
(E3 )
x ω 6 1 for x ∈ A∗
(E4 )
x ` ↔ x ω+` for x ∈ A∗ , ` > 0
(E5 )
x → x ` for x ∈ A∗ , ` > 0
(E6 )
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
ω
DECIDABLE Boolean algebra closed under quotients
18/20
Equations for u ∗
Pu =
(joint work with Charles Paperman)
[
u∗p
and
Su =
p prefix of u
[
su ∗
s suffix of u
x ω y ω = 0 for x , y ∈ A∗ such that xy 6= yx
(E1 )
∗
x y = 0 for x , y ∈ A such that y ∈
/ Px
(E2 )
yx ω = 0 for x , y ∈ A∗ such that y ∈
/ Sx
(E3 )
x ω 6 1 for x ∈ A∗
(E4 )
x ` ↔ x ω+` for x ∈ A∗ , ` > 0
(E5 )
x → x ` for x ∈ A∗ , ` > 0
(E6 )
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
ω
DECIDABLE Boolean algebra
18/20
The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6)
(u 6 )∗ u 2 ⊆ L if and only if (u 6 )∗ u 4 ⊆ L
19/20
The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6)
(u 6 )∗ u 2 ⊆ L if and only if (u 6 )∗ u 4 ⊆ L
19/20
The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
Equivalence relation over the integers
r ≡m s if and only if gcd(r , m) = gcd(s, m)
(u m )∗ u r ⊆ L if and only if (u m )∗ u s ⊆ L
2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6)
(u 6 )∗ u 2 ⊆ L if and only if (u 6 )∗ u 4 ⊆ L
19/20
The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
Equivalence relation over the integers
r ≡m s if and only if gcd(r , m) = gcd(s, m)
(u m )∗ u r ⊆ L if and only if (u m )∗ u s ⊆ L
2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6)
(u 6 )∗ u 2 ⊆ L if and only if (u 6 )∗ u 4 ⊆ L
19/20
The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
Equivalence relation over the integers
r ≡m s if and only if gcd(r , m) = gcd(s, m)
(u m )∗ u r ⊆ L if and only if (u m )∗ u s ⊆ L
x α ↔ x β for α and β representing sequences of
integers (km + r )k and (km + s)k with r ≡m s...
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The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
Equivalence relation over the integers
r ≡m s if and only if gcd(r , m) = gcd(s, m)
(u m )∗ u r ⊆ L if and only if (u m )∗ u s ⊆ L
d∗
b = {a}
x α ↔ x β for α and β profinite numbers in N
satisfying some specific conditions...
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The Boolean algebra
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
An example:
(a2 )∗ − (a6 )∗ = (a6 )∗ a2 ∪ (a6 )∗ a4
1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . .
Γ is the set of all the pairs of profinite numbers (dz P , dpz P ) s.t.:
P is a cofinite sequence of prime numbers {p1 , p2 , . . .}
z P = limn (p1 p2 . . . pn )n!
p∈P
if q divides d then q ∈
/P
x α ↔ x β for all (α, β) ∈ Γ
(E7 )
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Conclusion
Topology
Languages
Thank you for your attention
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