Groups( part 2):
Groupoid:
An oder pair a which the first element is a non empty set and the second element is a binary composition define on the set known as an algebraic structure/system.The algebraic system (G,*) is said to be a groupoid, if
Example.
Definitions.
A groupoid(G,*) is said to be a commutative groupoid, if the binary composition * is commutative.An element e in G is said to be an identity element in the groupoid(G,*) if a*e=e*a=a for all a in G.
For each element a in G if there exist an element a' in G s.t a*a'=a'*a=e then a' is called the inverse of a.
Note:
Semigroup.
A groupoid (G,*) is said to be a semigroup if * is associative.i) a*b in G for all a,b in G and
ii) a*(b*c)=(a*b)*c for all a, b, c in G .
Example.
(Z, -), (Q, +), (R, +) are semigroups.Monoid.
A semigroup (G, *) containing the identity element is said to be a monoid.i) a*b in G for all a,b in G,
ii) a*(b*c)=(a*b)*c for all a, b, c in G and
iii) there exists an element e in G such that e*a=a*e=a for all a in G.
Example.
(Z, +) is a monoid, 0 being the identity element.Quasigroup.
A groupoid (G,*) is said to be a quasigroup if for any two elements a,b in G, each of the equations a*x=b and y*a=b has a unique situation in G.Example.
(Z, +) is a groupoid. Let a,b in Z and the equation y+a=b has the solution y=b-a in Z. Therefore (z, +) is a quasigroup.Groups.
A non-empty set G is said to form a group with respect to a binary composition * , ifi) a*b in G for all a,b in G,
ii) a*(b*c)=(a*b)*c for all a, b, c in G,
iii) there exists an element e in G such that e*a=a*e=a for all a in G and
iv) for each element a in G, there exists an element a' in G such that a' *a=a*a'=e.
The group is denoted by the symbol (G, *).
The element e is said to be an identity element in the group.
The element a' is said to be an inverse of a.
The element a' is said to be an inverse of a.
For each element a in G if there exist an element a' in G s.t a*a'=a'*a=e then a'is called the inverse of a.
ReplyDelete