 Algebraic structure

An algebraic structure is a set equipped with specified algebraic operations satisfying selected axioms. An operation's domain is part of the structure: a binary operation on $S$ takes two elements of $S$ and produces another element of $S$. Groups specify one such operation, rings specify addition and multiplication, and vector spaces specify vector addition and scalar multiplication by a separate field.

Different choices of operations can give the same underlying set different structures. The integers form a group under addition, since every integer has an additive inverse. They do not form a group under multiplication: for example, $2$ has no multiplicative inverse among the integers. Even the nonzero integers fail that inverse requirement. In contrast, nonzero rational numbers form a group under multiplication.

Finite multiplication tables make such distinctions visible. Arithmetic modulo $5$ gives a field; arithmetic modulo $6$ gives a commutative ring with zero divisors. In the former, multiplying by any nonzero element permutes all residue classes. In the latter, multiplying by $2$ reaches only $0,2,4$, and $2\cdot 3=0$ even though both factors are nonzero.

The goal is to extract reusable properties rather than memorize isolated tables. A homomorphism preserves the specified operations, and an isomorphism shows that two structures have identical operation patterns after relabeling.

An algebraic structure is a set paired with one or more operations that satisfy certain axioms or rules. In mathematics, algebraic structures provide a framework for studying various mathematical concepts and properties. Here are some common types of algebraic structures: 1. **Groups**: A set \(G\) with a binary operation \(*\) that satisfies the following properties: - Closure: For all \(a, b \in G\), \(a * b \in G\).

A set $S$ plus any number of functions $f_i:S\times S\to S$, such that each $f_i$ satisfies some properties of choice.

Key examples: