binary operation二項演算にこうえんざん and closure閉包性へいほうせい

The first operation to check in abstract algebra抽象代数ちゅうしょうだいすう is an operation that takes two elements and produces one element. This is called a binary operation二項演算にこうえんざん.

A binary operation二項演算にこうえんざん on a set $$S$$S is a map

That is, for any $$a,b\in S$$a,b\in S, the result $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}b$$a\ast b is uniquely determined and is again an element of $$S$$S.

Here a map means a rule that assigns one output to each input. The set $$S\times S$$S\times S means the set of ordered pairs of elements of $$S$$S.

The condition that "the result returns to the same set" is closure閉包性へいほうせい. Thus closure is already built into the definition once we write $$\text{[PARSE ERROR: Undefined("Command("ast")")]}:S\times S\to S$$\ast:S\times S\to S. In practice, we often begin with a rule, such as addition or subtraction, that is meaningful on a larger domain and then test whether applying it to elements of $$S$$S always produces an element of $$S$$S.

1Why closure閉包性へいほうせい is necessary

Without closure閉包性へいほうせい, an operation cannot be repeated inside the same set.

For example, consider subtraction on the natural numbers $$\mathbb{N}$$\mathbb N:

The result is not a natural number. Therefore subtraction is not a binary operation二項演算にこうえんざん on $$\mathbb{N}$$\mathbb N.

On the other hand, if subtraction is considered on the integers $$\mathbb{Z}$$\mathbb Z, then for any integers $$a,b$$a,b, the result $$a-b$$a-b is an integer. Thus subtraction is a binary operation二項演算にこうえんざん on $$\mathbb{Z}$$\mathbb Z.

2Associative law and commutativity

The associative law結合法則けつごうほうそく says that for every $$a,b,c\in S$$a,b,c\in S, the position of parentheses does not affect the result when three or more elements are combined.

When the associative law結合法則けつごうほうそく holds, writing $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}b\text{[PARSE ERROR: Undefined("Command("ast")")]}c$$a\ast b\ast c is not ambiguous.

Commutativity可換性かかんせい says that for every $$a,b\in S$$a,b\in S, changing the order順序じゅんじょ does not change the result.

Commutativity is not always assumed. Matrix multiplication and composition of permutations are generally not commutative.

Matrices and permutations are mentioned only as examples of noncommutative operations; they are not used in the tests on this page.

3Identity element and inverse element逆元ぎゃくげん

An identity element単位元たんいげん is an element $$e\in S$$e\in S that leaves every $$a\in S$$a\in S unchanged on both sides.

Let $$e$$e be an identity element. An inverse element逆元ぎゃくげん of $$a\in S$$a\in S is an element $$a^{-1}\in S$$a^{-1}\in S that brings $$a$$a back to the identity in either order.

A two-sided identity, if it exists, is unique. Indeed, if both $$e$$e and $$f$$f are two-sided identities, then $$e=e\text{[PARSE ERROR: Undefined("Command("ast")")]}f=f$$e=e\ast f=f.

Identity elements and inverses are important for solving equations. To derive $$x=a^{-1}\text{[PARSE ERROR: Undefined("Command("ast")")]}b$$x=a^{-1}\ast b from $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}x=b$$a\ast x=b by operating with $$a^{-1}$$a^{-1} on the left and reassociating, assume associativity and an identity in addition to the inverse. Then

Thus applying $$a^{-1}$$a^{-1} on the left of both sides gives $$x=a^{-1}\text{[PARSE ERROR: Undefined("Command("ast")")]}b$$x=a^{-1}\ast b. Conversely, substituting this value gives $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}\left(a^{-1}\text{[PARSE ERROR: Undefined("Command("ast")")]}b\right)=\left(a\text{[PARSE ERROR: Undefined("Command("ast")")]}{a}^{-1}\right)\text{[PARSE ERROR: Undefined("Command("ast")")]}b=e\text{[PARSE ERROR: Undefined("Command("ast")")]}b=b$$a\ast(a^{-1}\ast b)=(a\ast a^{-1})\ast b=e\ast b=b, so it really is a solution.

Moreover, when associativity and a two-sided identity hold, a two-sided inverse of an element is unique if it exists. If $$u$$u and $$v$$v are both two-sided inverses of $$a$$a, then

The use of associativity in this argument is essential.

4Concrete example: reading an operation table

Put addition modulo 3 on the set $$S=\{0,1,2\}$$S=\{0,1,2\}: define $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}b$$a\ast b to be the remainder of $$a+b$$a+b upon division by 3.

The operation table is as follows, with the left operand indexing the row and the right operand indexing the column.

Every result lies in $$S$$S, so closure holds. The element 0 is the identity; the inverse of 0 is 0, the inverse of 1 is 2, and the inverse of 2 is 1. Both $$\left(a\text{[PARSE ERROR: Undefined("Command("ast")")]}b\right)\text{[PARSE ERROR: Undefined("Command("ast")")]}c$$(a\ast b)\ast c and $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}\left(b\text{[PARSE ERROR: Undefined("Command("ast")")]}c\right)$$a\ast(b\ast c) are the remainder of $$a+b+c$$a+b+c upon division by 3, so they are equal and the operation is associative. Thus this example is a group群ぐん, as defined in the next lecture.

5Exercise link

6Test supplement: check properties of an operation演算えんざん separately

Given a set $$S$$S and a rule $$\text{[PARSE ERROR: Undefined("Command("ast")")]}$$\ast, first check $$a,b\in S\Rightarrow a\text{[PARSE ERROR: Undefined("Command("ast")")]}b\in S$$a,b\in S\Rightarrow a\ast b\in S to see whether $$\text{[PARSE ERROR: Undefined("Command("ast")")]}:S\times S\to S$$\ast:S\times S\to S is a binary operation二項演算にこうえんざん. This checks only closure閉包性へいほうせい; it does not yet guarantee associativity, commutativity, an identity element単位元たんいげん, or inverse elements.

For example, subtraction $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}b=a-b$$a\ast b=a-b on $$\mathbb{Z}$$\mathbb Z has closure, but

so it does not satisfy associativity. Thus having closure and having group-like properties are separate questions.

7Summary

A binary operation二項演算にこうえんざん is a map that takes two elements of a set and produces an element of the same set. Closure閉包性へいほうせい, the associative law結合法則けつごうほうそく, commutativity可換性かかんせい, an identity element単位元たんいげん, and inverse elements逆元ぎゃくげん are the basic vocabulary used to define groups群ぐん, rings環かん, and fields体たい.