Monday, 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 1-1 / September 11, 2007

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 abc  G     (ab)a(bc)    
(2)<identity element>     e  G     arbitrary ∈ G     ae = ea = a     also expressed by 1G
(3)<inverse element>     e  G     arbitrary ∈ G     ab = ba = e      also expressed by a-1
Another additional condition of operation
(4)<commutative law>     arbitrary ab  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 ∈  ab  H
(2)∈  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
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 G
f (ab) = (af (b)

5
Definition of <isomorphism of group>
f is bijective, i.e. is injective ( map   B   a, a’  A     f a ) = a’  a = a’ ) and is surjective (map   B    Image ( ) = ).
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>     x  A  xe = ex = x
Definition of <commutative ring>
Another additional condition of operation
(4)<commutative law>    xy yx

Tokyo September 11, 2007
Sekinan Research Field of Language

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