Topic 5: Algebraic Structures
Operations on structures
there is a non-empty set called G. A function f from G × G to G is known as the binary operation on G. So f: G × G → G defines a binary operation on G.
Types of Algebraic Structures
Properties of Algebraic Structures
Associative
a, b, c in S, (a x b) x c = a x(b x c)
Identity
a x e = e x a = a
Closure
a, b in S, hence a x b in S.
Distributive
Commutive
a x b = b x a for every a, b Î S
Inverses
a x x = e.
Idempotent
What is Algebraic Structures
Algebraic structures are any formal mathematical systems consisting of a set of objects and operations on those objects.
Monoids
Semigroups
Groups
Abelian Group
the algebraic structure is a semigroup (S, ) if it is associative.commutative semigroup is a semigroup (S, ) at which * is a commutative operation.
monoid is a semigroup that has an identity
A Group is a monoid, but it contains an extra inverse element, which is denoted by 1. An algebraic structure (G, *) will be known as a group if it satisfies the following condition:
Associative
Inverses
Identity Element
An abelian group is a group, but it contains commutative law. An algebraic structure (G, *) will be known as an abelian group