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.
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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 )
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Definition of <isomorphism>
f is bijective.
f : A → B
Expression is A ≅ B
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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 }
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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.
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Definition of <residue class>
Group G
Normal subset of G H
Residue class is xH = Hx
Expression is x mod H
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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.
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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
Sekinan Research Field of Language
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