Q16Discrete Mathematics

Abstract algebra strictly categorizes mathematical structures based on the progressive accumulation of axiomatic properties. As a structure satisfies more axioms, it becomes mathematically richer and more constrained.

1. The Semigroup

A Semigroup is one of the most primitive algebraic structures. It consists of a non-empty set $S$ equipped with a single binary operation (let's denote it as $\ast$). To mathematically qualify as a Semigroup, the structure must satisfy exactly two strict axioms: 1. Closure: For any $a,b\in S$, the result of $(a\ast b)$ must also belong to $S$. 2. Associativity: For any $a,b,c\in S$, the grouping does not matter: $(a\ast b)\ast c=a\ast (b\ast c)$. An example is the set of even integers under addition. It is closed and associative, but it lacks the number 0 (the identity).

2. The Monoid

A Monoid is a direct mathematical upgrade to a Semigroup. A structure is a Monoid if it successfully satisfies all the axioms of a Semigroup (Closure and Associativity), while introducing a third, critical requirement: 3. Identity Element: There must exist a unique element $e\in S$ such that for every element $a\in S$, the equation $a\ast e=e\ast a=a$ holds perfectly true. An example is the set of all non-negative integers (Whole numbers) under addition. It possesses Closure, Associativity, and the number '0' acts as the perfect Identity element.

3. The Abelian Group

A Group is an upgrade to a Monoid, requiring every element to have a mathematical Inverse ($a\ast a^{-1}=e$). An Abelian Group (named after mathematician Niels Abel) is the ultimate, most heavily constrained structure in this hierarchy. It is a Group that also perfectly satisfies the Commutative property. Therefore, an Abelian Group must satisfy five absolute axioms: 1. Closure 2. Associativity 3. Identity 4. Inverse 5. Commutativity: For any $a,b$, the order is irrelevant: $a\ast b=b\ast a$. A classic example is the set of all Integers under addition. It has identity (0), inverses (negatives), and addition is flawlessly commutative.