Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 1-2 / September 12, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 1-2

    TANAKA Akio

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

8
Definition of <homomorphism of ring>
Ring     AB
Map    f : A  B
Arbitrary xy  A
y ) = f ( x ) + f ( )  and  f ( xy ) = 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 = { ( xy ) | x,y  X }
x R  x  y
Satisfied conditions are below.
(1) Reflective law     x ∼ x
(2) Symmetry law    ∼ y  ∼ x
(3) Transitivity law    ∼ y∼ ⇒ ∼ z
Definition of <equivalence class>
x  X
Subset of X    πx ) = { y  X | y  x }

11
Definition of <right coset>
Group     G
Subset      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 )
Definition of G / H
π π ( y ) =π xy )
G / is residue group.

14
Definition of <ideal>
Commutative ring    A
Subset    I  A
I is ideal by below conditions.
(1) xy  I  x – y  I
(2) x  Iy  A  xy = yx  I
Tokyo September 12, 2007
Sekinan Research Field of Language

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