Sunday, 1 April 2018

Linguistic Premise Premise of Algebraic Linguistics 1-3

Linguistic Premise

 Premise of Algebraic Linguistics 1-3

　　  TANAKA Akio

15
Definition of <presheaf>
Topological space     X
Arbitrary opened set     U, V    U < V
Commutative group     F  ( U )
Homomorphism     τUV : F  ( V ) → F  ( U )
Given conditions
(1)
F  ( 0 ) = { 0 }
(2)
τUV  = idU     (Identity map)
(3)
U ⊂ V ⊂ W     τUW  =τUV oτVV
Presheaf of commutative ring on X     { F ( U ), τUV }

16
Definition of <sheaf>
Presheaf F, G  on X
Homomorphism     ψ : F → G
Given conditions
(1)
Arbitrary opened set     U
Homomorphism     φ ( U ) : F ( U ) → G ( U )
(2)
Opened sets     U < V
Below makes commutative diagram.
F ( V ) , G ( V ), F ( U ) , G ( U )
φ ( V ), φ.( U )
τUV,
(3)
F  ( U ) ∋ s
Open covering U
{ U i }i ∈ I , r Ui, U = 0   i ∈ I   ⇒ s = 0
(4)
Open covering U
{ U i }i ∈ I
F ( U ) ∋ si     i ∈ I
rUi ∩Uj Ui (si ) = r Ui ∩Uj (sj) (i, j ∈ I )   ⇒ rUi, U (s) = si ( i ∈ I )
Presheaf is sheaf.

Tokyo September 17, 2007
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Linguistic Premise Premise of Algebraic Linguistics 1-2

Linguistic Premise

 Premise of Algebraic Linguistics 1-2

　　  TANAKA Akio

7
Definition of <subring>
Ring     A
Subset    B ∈ A
Identity element     1A ∈ B
x, y ∈ B ⇒ x-y ∈ B  and  xy ∈ B
B is subring.

8
Definition of <homomorphism of ring>
Ring     A, B
Map    f : A → B
Arbitrary x, y ∈ A
f ( x + y ) = f ( x ) + f ( y )  and  f ( xy ) = f ( x ) f ( y )

9
Definition of <isomorphism>
f is bijective.
f  : A → B
Expression is A ≅ B

10
Definition of <equivalence relation>
Set     X
Direct productive set     X × X = { ( x, y ) | x,y ∈ X }
( x, y ) ∈ R ⇔ x ∼ y
Satisfied conditions are below.
(1) Reflective law     x ∼ x
(2) Symmetry law    x ∼ y ⇒ y ∼ x
(3) Transitivity law    x ∼ y, y ∼ z ⇒ x ∼ z
Definition of <equivalence class>
x ∈ X
Subset of X    π( x ) = { y ∈ X | y ∼ x }

11
Definition of <right coset>
Group     G
Subset     H ∈ G
x ∼ y ⇔ x-1y ∈ H
Equivalence class of x ∈ G
Expression is xH.
Definition of <left coset>
x ∼ y ⇔ y x-1 ∈ H
Expression is Hx.

12
Definition of <residue class>
Group    G
Normal subset of G     H
Residue class is xH = Hx
Expression is x mod H

13
Definition of <residue group>
Group    G
Normal subset of G     H
Residue class    G / H
Map     π : G → G / H ; x → π ( x )
Definition of G / H
π ( x ) π ( y ) =π ( xy )
G / H is residue group.

14
Definition of <ideal>
Commutative ring    A
Subset    I ⊂ A
I is ideal by below conditions.
(1) x, y ∈ I ⇒ x – y ∈ I
(2) x ∈ I, y ∈ A ⇒ xy = yx ∈ I
Tokyo September 12, 2007

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Linguistic Premise Premise of Algebraic Linguistics 1-1

Linguistic Premise

 Premise of Algebraic Linguistics 1-1

　　  TANAKA Akio

1
Definition of <group>
Set     G   G ≠{0}
Operation     G × G → G ; ( a, b ) → ab    ab is called <product>.
Conditions of operation
(1) <associative law>     arbitrary a, b, c ∈ G     (ab)c = a(bc)    
(2)<identity element>     e ∈ G     arbitrary a ∈ G     ae = ea = a     also expressed by 1G
(3)<inverse element>     e ∈ G     arbitrary a ∈ G     ab = ba = e      also expressed by a-1
Another additional condition of operation
(4)<commutative law>     arbitrary a, b ∈ G     ab = ba                    G is called <Abelian group>.
When productive operation is done by <addition>, Abelian group is called <additive group> or <module>.

2
Definition of <subgroup>
Group     G
Subset H ∈ G
Conditions of H
(1)a, b ∈ H ⇒ ab ∈ H
(2)a ∈ H ⇒ a-1 ∈ H
Definition of <normal subgroup>
Arbitrary h ∈ H     g ∈ H
g-1 hg ∈ H
H     normal subgroup

3Definition of <finite group>
Group     G
G has finite elements.

G     finite group
Definition of <order>
G’s finite elements

4
Definition of <homomorphism of group>
Map     f : G → H
Arbitrary a, b ∈ G
f (ab) = f (a) f (b)

5
Definition of <isomorphism of group>
f is bijective, i.e. f is injective ( map  f : A → B   a, a’ ∈ A     f ( a ) = f ( a’ ) ⇒ a = a’ ) and f is surjective (map  f : A → B    Image ( f ) = B ).
Expression is G ≅ H

6
Definition of <ring>
Additive group     A
Product operation of A     AA → A ; ( x, y ) → xy
(1) <associative law>     (xy)z = x(yz)
(2) <distributive law>    (x + y)z = xz + yz    z(x + y) = zx + zy
(3) <identity element>   e ∈ A  x ∈ A  xe = ex = x
Definition of <commutative ring>
Another additional condition of operation
(4)<commutative law>    xy = yx

Tokyo September 11, 2007
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Algebraic Linguistics

Algebraic Linguistics
Subsite : sekinanlogos 

TANAKA Akio

<Being grateful to the mathematical pioneers>

On language universals, group theory is considered to be hopeful by its conciseness of expression. Especially the way from commutative ring to scheme theory is helpful to resolve the problems a step or two.

1 Linguistic Premise
2 Linguistic Note
3 Linguistic Conjecture
4 Linguistic Focus
5 Linguistic Result 

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Map 5 Analytic Theory

Map 5


Analytic Theory

TANAKA Akio

1 Algebraic Linguistics
2 Distance Theory Algebraically Supplemented
3 Noncommutative Distance Theory
4 Clifford Algebra
5 Kac-Moody Lie Algebra
6 Operator Algebra
7 von Neumann Algebra
8 Functional Analysis
9 Reversion Analysis Theory
10 Holomorphic Meaning Theory
11 Stochastic Meaning Theory
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Theory Map

Sekinan Linguistic Field

We cannot get any deep understanding of the laws that govern the physical world without entering the world of mathematics.
Roger PENROSE, THE ROAD TO REALITY

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Quantum Linguistics Creation of Meaning Toward Realization of Quantum Language Machine 2


Quantum Linguistics

Creation of Meaning
Toward Realization of Quantum Language Machine 2

TANAKA Akio

A
1 In quantum linguistics, language consists of quanta.
2 Quantum simultaneously has frame and transmission† in folded condition.
3 Frame and transmission correspond to solidity and flexibility of language.
4 Frame is used for entry letters.
5 Transmission is used for description.
6 From physical viewpoint, language consists of quantum bits.
7 One quantum bit expresses folded two conditions. Some conditions are used for frame and the others are for transmission.
8 The remained conditions of folding are used for description.
B
9 One of main theme†† for description is meaning that is, ideally, recognized by quantum itself.
10 For example, 4 quantum bits express 16 folded conditions, namely 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, and 1111.
11 For example, word “be” of natural language, abbreviated to NL, is expressed by quantum bits as following.
12 Word “be” is translated to quantum language’s meaning *existence*. *   * is the notation for meaning of QL.
13 *Existence* is expressed as existence of a set in quantum language machine.
14 Now zero set, expressed by φ, is used for a set of this case. Zero set is defined to {x | x ≠x}. So *existence* is expressed to {φ}.
15 Word “be” is described by 4 quantum bits as following. b0000, e0001, {0010, {0011, x0100, |0101, x0110, ≠0111, x1000, }1001, }1010. Remained conditions are not used immediately.
16 Next, word “not” of natural language is expressed by quantum bits as following.
17 Word “not” is translated to QL *vacancy*.
18 *Vacancy* is expressed toφitself. Bits expression is omitted
19 Words connection “be not” is expressed to {φ}φ. Bits expression is omitted.
20 *Existence* is also means *1* in QL. So *2* is expressed to {φ, 1}. Following numbers are omitted.
21 For example, Sentence of NL, “There is an apple.” is expressed as following ideally. {φ} *apple* {φ}. Here *apple* has not been recognized by QLM.


[Postscript March 14, 2008]
† On <frame and transmission>, refer to the next.
†† On the theme of this paper, more details are below.
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