Basic definitions

Basic definitions
Definition 1.1 : The Cartesian product of two sets A and B, distinct or
not, is the set *( ) +.
Definition 1.2 : If , then there exist unique integer q
and r such that .
Definition 1.3: Let a and b be integers, not both of which are zero. The
greatest common divisor of a and b , denoted by ( ), is positive
integer d such that
1) d\a and d\b
2) if c\a and c\b then c\d.
Definition 1.4: The least common multiple of two nonzero integers a
and b , denoted by ( ), is positive integer e such that
1) a\e and b\e
2) if a\c and b\c then e\c.
Definition 1.5:Given a nonempty set S, any function from Cartesian
product into S is called a binary operation on S.
Definition 1.6: The operation defined on the set S is said to be
associated if ( ) ( ) for every triple distinct or not of
elements a, a, and c of S.
Definition 1.7: A semigroup is a pair ( )consisting of a nonempty set S
together with an associative( binary) operation defind on S.
Definition 1.8: The operation defind on S is called commutative if
for every pair of elements .
Definition 1.9: Two elements a and b are said to be commute or
permute with each other provided .
Definition 1.10: The semigroup ( ) is said to have a (two-sides) identity
element for the operation if there exist an element e in S such that
. An element e having this property is called
an identity element ( unit element, neutral element ) for ( ).
Theorem 1.11: A semigroup ( ) has at most one identity element.
Proof: Suppose that ( ) has two elements identity . By
definition of identity
, then in particular
. ?
Definition 1.12: Let ( ) semigroup with identity ( e ). An element
is said to have a (two-sides) inverse under operation if there
exist an element in S such that .
Theorem 1.13: A semigroup ( ) with identity has at most one inverse .
Proof: Suppose that the element a in ( ) has two inverses elements
. By definition of identity
, then
, thus
then
. ?
Definition 1.14: The pair ( ) is a group if and only if ( ) is a
semigroup with identity, in which every element in G has an inverse.
Definition 1.15: A group is a pair ( ) consisting of a nonempty set G
and a binary operation defined on G, satisfying the four requirements:
1) G is closed under operation .
2) The operation is associative.
3) G contains an identity element e.
4) each element has an inverse .