Algebraic Structures and Coding
Theory
Algebraic system
-Algebraic system is defined as, it is an ordered pair (A,),where A is the set of
elements and * is set of m-ary operations such that ɏ a, b ɛ A, ab ɛ A
-Note that * is not multiplication if the set of m-ary operations like (A,+),(A,+,) etc
-Foe example Let E={0,2,4,…} then set E along with binary operation ‘+’(addition) is
algebraic system
-Set of integers Z with operations ‘+’ and ‘’ is the algebraic system i.e. (Z,+,*)
Properties of Binary Operations
1. Commutative properties:- A binary operation‘’ is called as commutative if
ab=b*a for all elements of a,b ɛ A.
eg.
2. Binary operation addition on integers is commutative, but substraction is not
commutative
3. Binary operation multiplication on the set of integer is commutative
Associative Property
-A binary operation ‘’ is called as associative if a(bc)=(ab)*c for all elements of a,b,c
ɛ A.
• For example
1. Binary operation addition is associative, but operation subtraction is not
associative on set of integers.
2. Binary operation multiplication is associative on the set of integer.
Idempotent Property
Algebraic Structures and Coding Theory 1
, -A binary operation * on set of a is called as idempotent if it satisfy the property
a*a=a for all a ɛ A
-For example Consider a Lattice L with two operations meet and join .then ᴧ and v
operations such that for all a ɛ A a ᴧ a= a and a v a= a. Therefore both ᴧ and v
satisfy idempotent law
Identity Property
An element e ɛ A is called as identity element for the operation * ,if ae =ea=a for all
aɛA
Distributive Property
For a,b,c ɛ A and the operation * and + the operation ‘’ distributes over the ‘+’ If
a(b+c)=(ab)+(ac). Similarly operation + distributes over * If a+(bc)= (a+b)(a+c)
Cancellation Law
For a,b,c ɛ A and operation satisfy the cancellation law (property) then ab=ac=b=c
Diverse Element
for each element a ɛA, there exist an element b ɛA such that a* b=e where e is
identity element
Semi-Group
An algebraic system (A,) with a binary operation * on A is said to be a semigroup if *
is associative a(bc)=(ab)c for all a,b,c ɛ A
e.g.
1)if R is set of real numbers then(R,+) and (R,) is semigroup.this is also
commutative therefore it is also called commutative semigroup
2)Z is set of integers and (Z,+) is commutative semigroup
3)(Z,-) is not semigroup as subtraction operation is not associative
Monoid
-A semi group (A,) with an identity element is called as monoid.
-Example
1)(z,+) is a monoid where o is an identity element is called as monoid. 7+0=0+7=7
2)(z,) is a monoid where 1 is the identity element 71=17=7
Group
Algebraic Structures and Coding Theory 2
Algebraic Structures and Coding
Theory
Algebraic system
-Algebraic system is defined as, it is an ordered pair (A,),where A is the set of
elements and * is set of m-ary operations such that ɏ a, b ɛ A, ab ɛ A
-Note that * is not multiplication if the set of m-ary operations like (A,+),(A,+,) etc
-Foe example Let E={0,2,4,…} then set E along with binary operation ‘+’(addition) is
algebraic system
-Set of integers Z with operations ‘+’ and ‘’ is the algebraic system i.e. (Z,+,*)
Properties of Binary Operations
1. Commutative properties:- A binary operation‘’ is called as commutative if
ab=b*a for all elements of a,b ɛ A.
eg.
2. Binary operation addition on integers is commutative, but substraction is not
commutative
3. Binary operation multiplication on the set of integer is commutative
Associative Property
-A binary operation ‘’ is called as associative if a(bc)=(ab)*c for all elements of a,b,c
ɛ A.
• For example
1. Binary operation addition is associative, but operation subtraction is not
associative on set of integers.
2. Binary operation multiplication is associative on the set of integer.
Idempotent Property
Algebraic Structures and Coding Theory 1
, -A binary operation * on set of a is called as idempotent if it satisfy the property
a*a=a for all a ɛ A
-For example Consider a Lattice L with two operations meet and join .then ᴧ and v
operations such that for all a ɛ A a ᴧ a= a and a v a= a. Therefore both ᴧ and v
satisfy idempotent law
Identity Property
An element e ɛ A is called as identity element for the operation * ,if ae =ea=a for all
aɛA
Distributive Property
For a,b,c ɛ A and the operation * and + the operation ‘’ distributes over the ‘+’ If
a(b+c)=(ab)+(ac). Similarly operation + distributes over * If a+(bc)= (a+b)(a+c)
Cancellation Law
For a,b,c ɛ A and operation satisfy the cancellation law (property) then ab=ac=b=c
Diverse Element
for each element a ɛA, there exist an element b ɛA such that a* b=e where e is
identity element
Semi-Group
An algebraic system (A,) with a binary operation * on A is said to be a semigroup if *
is associative a(bc)=(ab)c for all a,b,c ɛ A
e.g.
1)if R is set of real numbers then(R,+) and (R,) is semigroup.this is also
commutative therefore it is also called commutative semigroup
2)Z is set of integers and (Z,+) is commutative semigroup
3)(Z,-) is not semigroup as subtraction operation is not associative
Monoid
-A semi group (A,) with an identity element is called as monoid.
-Example
1)(z,+) is a monoid where o is an identity element is called as monoid. 7+0=0+7=7
2)(z,) is a monoid where 1 is the identity element 71=17=7
Group
Algebraic Structures and Coding Theory 2
