Algebraic Structures

Overview

An algebraic structure is a nonempty set $A$ , a collection of operations on $A$ , and a finite number of axioms these operations must satisfy.

Substructures

Let $S$ be an algebraic structure and $T$ be a subset of the corresponding set of $S$ , along with the naturally induced operations of $S$ . If $T$ is an algebraic structure of the same kind as $S$ , we say $T$ is a substructure of $S$ .

This is denoted as $T \leq S$ . If $T \leq S$ and $T \neq S$ , then we can state $T < S$ .

Direct Products

The direct product of a collection of algebraic structures is a structure of the same type combined with a component-wise analogue of the structures' operations. The direct product of structures $S_{1}, S_{2}, \dots, S_{n}$ is denoted as

⨂_{i = 1}^{n} S_{i} = S_{1} \times S_{2} \times \dots \times S_{n} .

It's underlying set is the Cartesian product of the underlying sets of the given structures.

Direct Sums

If the operation associated with a direct product is commutative, we sometimes prefer additive notation. In this case, we instead use the term direct sum and denote the product as:

⨁_{i = 1}^{n} S_{i} = S_{1} \oplus S_{2} \oplus \dots \oplus S_{n} .

Internal/External

An algebraic structure $S$ is the internal direct product of substructures $T_{1}, T_{2}, \dots, T_{n}$ if and only if

Every $s \in S$ is uniquely expressed as a product $s = t_{1} t_{2} \dots t_{n}$ where $t_{i} \in T_{i}$ , and
Elements of different substructures commute with one another.

In contrast, the previous definition of a direct product is called the external direct product.

Magmas

A magma (or binary algebraic structure) is a set $M$ equipped with a single binary operation on $M$ .

Identity Elements

Let $⟨ S, * ⟩$ be a magma. An element $e \in S$ is an identity element for $*$ if $e * s = s * e = s$ for all $s \in S$ . A magma has at most one identity element. That is, if there is an identity element, it is unique.

Semigroups

A semigroup is a magma whose binary operation is associative.

Monoids

A monoid is a semigroup with an identity element.