ABSTRACT ALGEBRA:
EXPLORING FUNDAMENTAL
ALGEBRAIC STRUCTURES
AND APPLICATIONS
,INTRODUCTION
Abstract algebra is a branch of mathematics
that studies algebraic structures in a more
generalized and abstract way. It focuses on
the fundamental concepts and properties
that apply to a wide range of mathematical
systems. This note provides an overview of
various topics in abstract algebra.
, SEMICROUPS AND
MONOIDS
Semigroups:
A semigroup is a set equipped with an
associative binary operation. Formally, a
semigroup (S, *) consists of a set S and a
binary operation *: S x S -> S such that for
all a, b, and c in S, the operation * satisfies
the associative law: (a * b) * c = a * (b * c).
Example: Let S be the set of positive
integers and * be the operation of addition.
The set S, together with the operation *,
forms a semigroup. For example, (2 + 3) +
4 = 9 and 2 + (3 + 4) = 9, satisfying the
associative law.
Monoids:
A monoid is a semigroup with an identity
element. A monoid (M, *) consists of a set M
and a binary operation *: M x M -> M, which
is associative, and an identity element e in
M such that for every element a in M, the
following holds: a * e = e * a = a.
Example: Let M be the set of non-negative
integers and * be the operation of
EXPLORING FUNDAMENTAL
ALGEBRAIC STRUCTURES
AND APPLICATIONS
,INTRODUCTION
Abstract algebra is a branch of mathematics
that studies algebraic structures in a more
generalized and abstract way. It focuses on
the fundamental concepts and properties
that apply to a wide range of mathematical
systems. This note provides an overview of
various topics in abstract algebra.
, SEMICROUPS AND
MONOIDS
Semigroups:
A semigroup is a set equipped with an
associative binary operation. Formally, a
semigroup (S, *) consists of a set S and a
binary operation *: S x S -> S such that for
all a, b, and c in S, the operation * satisfies
the associative law: (a * b) * c = a * (b * c).
Example: Let S be the set of positive
integers and * be the operation of addition.
The set S, together with the operation *,
forms a semigroup. For example, (2 + 3) +
4 = 9 and 2 + (3 + 4) = 9, satisfying the
associative law.
Monoids:
A monoid is a semigroup with an identity
element. A monoid (M, *) consists of a set M
and a binary operation *: M x M -> M, which
is associative, and an identity element e in
M such that for every element a in M, the
following holds: a * e = e * a = a.
Example: Let M be the set of non-negative
integers and * be the operation of
