Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 3-1 / September 23, 2007


Linguistic Premise

 Premise of Algebraic Linguistics 3-1

    TANAKA Akio

1 <finitely generated>
Group     G
Subset of G     S
G is generated by finite set S.     G is finitely generated.

2 <ascending chain rule>
Commutative ring     A
Ideal of A   
Ascending chain of stops finitely.
The situation satisfies ascending chain rule.

3 <descending chain rule>
Commutative ring     A
Ideal of A   
Descending chain of stops finitely.
The situation satisfies descending chain rule.


4 <maximum element>
Set defined by order     X
Element of X     ax
x that is < x does not exist.
a is maximum element.

5 <minimum element>
Set defined by order     X
Element of X     bx
x that is > x does not exist.
b is minimum element.


4 <Noetherian ring>
Commutative ring A that satisfies next equivalent conditions is Noetherian ring.
(1) A satisfies ascending chain rule on ideal.
(2) Ideal family of A has maximum element.
(3) Ideal of A is finitely generated.

5 <Artinian  ring>
Commutative ring A that satisfies next equivalent conditions is Artininian ring.
(1) A satisfies descending chain rule on ideal.
(2) Ideal family of A has minimum element.

6 <module>
Additive group     M
Ring     A
M that has action of is A module.
M satisfies next conditions.
(1) a ( x + y ) = ax ay,     ( a + b ) x = ax + bx
(2) bx ) = ( ab ) x,     1x = x

7 <direct sum>
A module     MN
Structure of A module is given by set of product  M ×N. The situation is expressed by M  N.
Direct sum of n-M, i.e. M M ….M is expressed by Mn.

Tokyo September 23, 2007
Sekinan Research Field of Language

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