Algebraic Structures

An algebraic structure is a mathematical framework that consists of a non-empty set \( A \), whose elements can be of any type, provided that one or more operations are defined on this set.

Operations within an algebraic structure can be either internal, meaning the results of operations on elements of \( A \) remain within \( A \), or external, involving elements from other sets.

The nature of these operations and their respective properties dictate the kind of algebraic structure in question. For instance:

• Groupoids: A basic algebraic structure consisting of a set paired with a closed binary operation. Every operation within this set yields another set element.
• Semigroups: An extension of the groupoid that incorporates associativity, ensuring the order of operations does not alter the outcome.
• Monoids: Further building on the semigroup, this structure adds an identity element that does not affect the outcome of operations involving other elements.
• Groups: A set \( A \) paired with an internal binary operation that adheres to closure, associativity, identity, and invertibility.
• Rings: A set \( A \) equipped with two binary operations (commonly addition and multiplication) that respect closure, associativity for both operations, distributivity of multiplication over addition, and an identity element for addition.
• Fields: These are special types of rings where both operations (addition and multiplication) are commutative, and every non-zero element has a multiplicative inverse.

The set \( A \) is referred to as the carrier of the structure, and the combination of the carrier and its defined operations constitutes the algebraic structure itself.

Each algebraic structure is governed by specific rules that regulate the behavior of its operations and influence the properties of the set.

Example

Consider the set of natural numbers \( \mathbb{N} \), which encompasses all non-negative integers \( \{0, 1, 2, 3, \ldots\} \).

We define the addition operation \( + \) on this set.

Let's explore how \( (\mathbb{N}, +) \) forms a particular algebraic structure, specifically a commutative monoid.

It satisfies the following properties:

1. Closure: The sum of any two natural numbers results in another natural number.

   For example, \( 2 + 3 = 5 \) and \( 5 \in \mathbb{N} \).

2. Associativity: Addition is associative, meaning that the sum of any three natural numbers \( a \), \( b \), and \( c \) can be grouped in any order without changing the result.

   For example: $$ (2 + 3) + 4 = 9 $$ and $$ 2 + (3 + 4) = 9 $$, thus $$ (2 + 3) + 4 = 2 + (3 + 4) $$

3. Identity Element: Zero (\( 0 \)) acts as the identity element for addition, as adding zero to any number \( a \) in \( \mathbb{N} \) leaves it unchanged.

   For instance, \( 5 + 0 = 5 \).

4. Commutativity: Addition is commutative, meaning the order of addition does not affect the result; \( a + b = b + a \) for any \( a \) and \( b \) in \( \mathbb{N} \).

   For example, \( 3 + 4 = 4 + 3 \).

While natural numbers with addition form a commutative monoid, they do not constitute a group because they lack the invertibility property for each element.

For example, there is no natural number which, when added to 5, yields 0. That is, addition in \( \mathbb{N} \) does not always provide an additive inverse. The only natural number with an inverse element is 0, which inversely is itself, as \( 0+0=0 \).

This algebraic structure serves as an essential foundation in mathematics and is pivotal in various fields like number theory and computer science.