4 Algebraic System

8/17/2019 4 Algebraic System

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ALGEBRAIC SYSTEMS


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Binary OperationSuppose G is a nonempty set. Operations on the

elements of a set is a rule, which combines twoelements of the set to obtain a third of the set.

When on performing this operation on any two

elements of the set, we obtain a unique element

belonging to the same set, then this operation is

called binary operation, on the set.

Ex. !,"#

$hus %&' will be a binary operation on G, iff

.&

,,&,

uniqueisbawhere

GbaGbaGbGa ∈∀∈⇒∈∈


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Algebraic structure

• A non empty set G equipped with one or more

binary operation is called an algebraic structure.

• Groupoid - Groupoid is the simplest algebraic

structure using only one composition(G,*).

Semigroup

•

A non empty set G equipped with an associativebinary operation ( say !).

• "epresented by (G , )

• #$amples % ( &, ' )


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onoid

• a semi group G ,∗# is said to be a monoid if G

possess identity element with respect to %&'.

Example (

), " # with e * +, S#,∪ # with e *φ

•

E-ery monoid is a semigroup but e-ery semigroupmay not be a monoid.


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Group

• a set of elements or numbers/ with someoperation whose result is also in the set closure#

•

obeys( – associati-e law((a.b).c = a.(b.c)

– has identity e( e.a = a.e = a

– has in-erse a-1(a.a-1 = e

• if commutati-e a.b = b.a – then forms an abelian group


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Group

e.g.

{ }

#,

,0,1,+,12,02,*

3additionarithmeticintegers,of setthe4*1

+S

S

G

5losureyes, 6"

"addition, yes, 6 Op.Binary

∴∈

=

S ba

( ) ( ) e7ssociati-yes, 6""*"" ∴cbacba

Element)n-ersehas yes, 6 +"

+*eyes, 6 *"

12∴=

∴

aa

aea


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Group ounter-#$ample

#,

3tionmultiplicaintegers,of setthe4

0∗=

=

Z

G

5losure yes, 6

tion,multiplica yes, 6 op binary

∴∈∗

∗=

S ba

( ) ( )

1*eyes, 6 *

e7ssociati-yes, 6*

∴∗

∴∗∗∗∗

aea

cbacba

groupanottrue,isthisfor whichnoisthere !o, 61

1212

∴

=∗ aaa


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roperties o+ Groups

• $he identity element is unique.

• $he in-erse of each element is unique.

• )f in-erse of a is a21, then the in-erse of a21 is a

i.e., a21

#21

* a.• $he in-erse of the product of two elements of a

group is the product of the in-erses ta8en in there-erse order i.e., ab#21 * b21a21

•

5ancellation laws hold i.e., ab * ac ⇒ b * c 9eft cancellation law #

ba * ca ⇒ b * c right cancellation law #


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et e,e! two identity elements o+ Group (G, )

then ee' * e if e' is identity.

and ee' * e' if e is identity.

but ee! is unique element o+ G

ence e e!

/he identity element in a group is unique.

roperties o+ Groups


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Group• 7belian group ( a group G ,∗# is abelian

if ∗ is commutati-e.• Example( ) , ∗ # where ) is the set of

integers and operation is defined as

a∗ b * a"b20 for all a,b in )i. a∈), b∈) ⇒ a"b20∈) so ) is closed w.r.t.∗

ii. a∗ b#∗c * a ∗b ∗ c# 7ssociati-e#

a∗ b#∗c * a"b20#∗c * a"b:0#"c20 *a"b"c2;a∗b∗c# *a ∗b"c20#*a"b"c20#20 * a"b"c2;


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Group #$ample

iii. )dentity

e∗a * a

e"a20 * a ⇒ e * 0∈) for all a in )

i-. )n-erse

)f a∈) then b∈) will be the in-erse of a if

a∗ b * e* b∗a

a"b20 *0 ⇒ b* 2a"; ∈)

– )s ) an abelian group?


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omomorphism o+ semigroups•

)f G , ∗ # and G' , ∗' # be twosemigroups,a mapping f( G→G' is called asemigroup homomorphism or ahomomorphism if

f a∗ b# * fa# ∗' fb#


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#$ample

• 9et f (


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0"1#"

• Order of an element of a group

• 9et G ,∗# be a group, a ∈ G and e is

identity .Order of an element a is the least positi-e integer n such that

• a∗a ∗a ∗a ∗= ∗ a n times# * eidentity#

or • an * e if %∗' is %x'

• na * e if %∗' is %"'


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7ddition and >ultiplication modulo• 7ddition modulo n

a"n b * r, n > r ≥ +• r * least nonnegati-e remainder when a"b is

di-ided by n.

•

example ( 10"?@ * 1, 01"11?+* @• >ultiplication modulo m

• a×m b * r, m > r ≥ +

• r * least nonnegati-e remainder when a× bis di-ided by m.

• Example( 10×A * 1, 1?×@A * 0


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>odulo Example

composition table for addition modulo


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>odulo A Example

composition table for multiplication modulo A

×A

1 0 ? ;

1 1 0 ? ;

0 0 ; 1 ?

? ? 1 ; 0

; ; ? 0 1


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>odulo Example

composition table for addition modulo

• Exercise 1. Write the composition table for

addition modulo C.

• Exercise 0.Show that the set 41,?,;,A,D3 isan abelian group under multiplication

modulo 11 as composition.


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5yclic Group

•

7 group is cyclic if e-ery element is a power of some fixed element

b = ak

+or some a and every b ingroup

•

a is said to be a generator of the group


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e.g. et G23,-3,i,-i4 is a group with respect to the binary

operation ./hen G is a cyclic group. 5ind the

generators o+ a group G.Ans. i and 6i.

e.g. Show that is cyclic. ow many

generators are there 7

×

#3,C,A,;,?,0,14 @×


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5yclic Group

• E-ery cyclic group is an abelian group.

• )f a finite group of order n contains an

element of order n, the group must be

cyclic.


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)E9F

$he system ,",.# is a field if,

12 ,"# is an abelian group.

02 +,.# is an abelian group.

?2 >ultiplication is distributi-e w.r.t addition.O

4 Algebraic System

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