Chapter 2: Mathematical Preliminaries ~ 39
We give below a few examples of sets with one binary operation:
(i) Z with addition is an abelian group.
(ii) Z with multiplication is an abelian monoid. (It is not a group since it
does not satisfy the postulate 4.)
(iii) {I, 2. 3, ... } with addition is a commutative semigroup but not a
monoid. (The identity element can be only 0, but 0 is not in the set.)
(iv) The power set 2
4 of A(A -j; 0) with union is a commutative monoid.
(The identity element is 0.)
(v) The set of all 2 x 2 matrices under multiplication is a monoid but not
an abelian monoid.
2.1.3 SETS WITH Two BINARY OPERATIONS
Sometimes we come across sets with two binary operations defined on them
(for example, in the case of numbers we have addition and multiplication). Let
5 be a set with two binary operations * and o. We give below 11 postulates
in the following way:
(i) Postulates 1-5 refer to * postulates.
(ii) Postulates 6. 7. 8. 10 are simply the postulates L 2. 3, 5 for the binary
operation o.
(iii) Postulate 9: If 5 under 8 satisfies the postulates 1-5 then for every x
in S. with x -j; e, there exists a unique element x' in 5 such that x' 0 x =
x 0 x' = e ' , where e ' is the identity element corresponding to o.
(iv) Postulate 11: Distributivil\'. For a. b. c. in 5
a 0 (b * c) = (a 0 b) * (a 0 c)
A set with one or more binary operations is called an algebraic system.
For example, groups, monoids, semigroups are algebraic systems with one
binary operation,
We now define some algebraic systems with two binary operations.
Definitions (i) A set \vith two binary operations * and 0 is called a ring if
(a) it is an abelian group W.f.t. 8, and (b) 0 satisfies the closure, associativity
and distributivity postulates (i.e. postulates 6. 7 and 11).
(ii) A ring is called a commutative ring if the commutativity postulate is
satisfied for o.
(iii) A commutative ring with unity is a commutative ring that satisfies the
identity postulate (i,e. postulate 8) for o.
(iv) A field is a set with two binary operations * and 0 if it satisfies the
postulates 1-11.
We now give below a few examples of sets with two binary operations:
(i) Z with addition and multiplication (in place of * and 0) is a
commutative ring with identity. (The identity element W.f.t. addition is
O. and the identity element \V.r.t. multiplication is 1.)
We give below a few examples of sets with one binary operation:
(i) Z with addition is an abelian group.
(ii) Z with multiplication is an abelian monoid. (It is not a group since it
does not satisfy the postulate 4.)
(iii) {I, 2. 3, ... } with addition is a commutative semigroup but not a
monoid. (The identity element can be only 0, but 0 is not in the set.)
(iv) The power set 2
4 of A(A -j; 0) with union is a commutative monoid.
(The identity element is 0.)
(v) The set of all 2 x 2 matrices under multiplication is a monoid but not
an abelian monoid.
2.1.3 SETS WITH Two BINARY OPERATIONS
Sometimes we come across sets with two binary operations defined on them
(for example, in the case of numbers we have addition and multiplication). Let
5 be a set with two binary operations * and o. We give below 11 postulates
in the following way:
(i) Postulates 1-5 refer to * postulates.
(ii) Postulates 6. 7. 8. 10 are simply the postulates L 2. 3, 5 for the binary
operation o.
(iii) Postulate 9: If 5 under 8 satisfies the postulates 1-5 then for every x
in S. with x -j; e, there exists a unique element x' in 5 such that x' 0 x =
x 0 x' = e ' , where e ' is the identity element corresponding to o.
(iv) Postulate 11: Distributivil\'. For a. b. c. in 5
a 0 (b * c) = (a 0 b) * (a 0 c)
A set with one or more binary operations is called an algebraic system.
For example, groups, monoids, semigroups are algebraic systems with one
binary operation,
We now define some algebraic systems with two binary operations.
Definitions (i) A set \vith two binary operations * and 0 is called a ring if
(a) it is an abelian group W.f.t. 8, and (b) 0 satisfies the closure, associativity
and distributivity postulates (i.e. postulates 6. 7 and 11).
(ii) A ring is called a commutative ring if the commutativity postulate is
satisfied for o.
(iii) A commutative ring with unity is a commutative ring that satisfies the
identity postulate (i,e. postulate 8) for o.
(iv) A field is a set with two binary operations * and 0 if it satisfies the
postulates 1-11.
We now give below a few examples of sets with two binary operations:
(i) Z with addition and multiplication (in place of * and 0) is a
commutative ring with identity. (The identity element W.f.t. addition is
O. and the identity element \V.r.t. multiplication is 1.)