What is an algebraic structure代数的構造だいすうてきこうぞう?

The first viewpoint to fix in abstract algebra抽象代数ちゅうしょうだいすう is to regard an object as a pair consisting of a set集合しゅうごう and operations演算えんざん. Algebra does not begin from a set of numbers alone. Only after specifying which operations are performed inside that set and which laws those operations satisfy do we obtain an algebraic structure代数的構造だいすうてきこうぞう.

For example, the set of all integers $$\mathbb{Z}$$\mathbb Z becomes a group群ぐん when viewed with addition. But with multiplication it is not a group: for example, the only candidate satisfying $$2b=1$$2b=1 is $$b=1/2$$b=1/2, which is not an integer. Even with the same underlying set, changing the operation changes the structure.

The terms group群ぐん and inverse element逆元ぎゃくげん will be defined formally later. At this point, we use only the idea that a structure is not determined by a set alone; the operation must also be specified.

1The form of a definition定義ていぎ

An algebraic structure代数的構造だいすうてきこうぞう is typically written in a form such as

or

Here $$S$$S is a set集合しゅうごう, and $$\text{[PARSE ERROR: Undefined("Command("ast")")]}$$\ast, $$+$$+, and $$·$$\cdot are operations.

This notation separates "what the elements are" from "how the elements are operated on." A set alone is only a list or collection of elements, and an operation alone does not say where the calculation takes place. Only when both are specified is the world of calculation determined.

2Why define by axioms?

Axioms公理こうり are conditions that let us forget the details of concrete examples and keep only the properties needed for an argument.

Addition of integers, rotations of the plane, and symmetries of a square look completely different. However, they share common features: operations can be composed, there is an operation that does nothing, and there are operations that return an element to where it was. Extracting just these common features gives the axioms of a group群ぐん.

3What changes and what is preserved

In abstraction, the concrete presentation changes. Integers, matrices, permutations, and residue classes look different. On the other hand, we preserve and study the laws satisfied by the operations.

For example, two isomorphic groups may have elements with different names, but the structure of their operation tables is the same. In other words, the information about which element is obtained by combining which elements is preserved. Isomorphism will be defined later together with homomorphisms, so here it should be read only as the preview that operation structure can remain the same even when names change.

4Concrete example: the same set can have different structures

Consider the set $$\mathbb{Z}$$\mathbb Z.

is a group群ぐん. The identity element単位元たんいげん is 0, and the inverse of an integer $$a$$a is $$-a$$-a.

However,

is not a group. For example, the multiplicative inverse of 2 does not exist among the integers. If there were an integer $$b$$b with $$2b=1$$2b=1, the left-hand side would be even, so it could not equal 1.

The important point is that we cannot decide whether something is a group by looking only at the set. We must always judge it together with its operation.

5Connections with other areas

abstract algebra抽象代数ちゅうしょうだいすう also appears in linear algebra. For example, the set of all invertible matrices forms a group群ぐん under matrix multiplication.

This section is only a preview of connections; matrix knowledge is not used as a prerequisite for later definitions or proofs here.

In number theory, addition and multiplication of residue classes become basic examples in abstract algebra抽象代数ちゅうしょうだいすう.

6Exercise link

After this lecture, continue to binary operations and closure, then to semigroups, monoids, and groups.

7Summary

An algebraic structure代数的構造だいすうてきこうぞう is a set equipped with operations, with required properties specified by axioms. Even if the underlying set is the same, changing the operation changes the structure. In abstract algebra抽象代数ちゅうしょうだいすう, we focus not on appearance but on the structure preserved by operations.