# Abstract Algebra/Binary Operations

A binary operation on a set $A$ is a function $*:A\times A\rightarrow A$. For $a,b\in A$, we usually write $*(a,b)$ as $a*b$.

## Properties

The property that $a*b\in A$ for all $a,b\in A$ is called closure under $*$.
Example: Addition between two integers produces an integer result. Therefore addition is a binary operation on the integers. Whereas division of integers is an example of an operation that is not a binary operation. $1/2$ is not an integer, so the integers are not closed under division.
To indicate that a set $A$ has a binary operation $*$ defined on it, we can compactly write $(A,*)$. Such a pair of a set and a binary operation on that set is collectively called a binary structure. A binary structure may have several interesting properties. The main ones we will be interested in are outlined below.
Definition: A binary operation $*$ on $A$ is associative if for all $a,b,c\in A$, $(a*b)*c=a*(b*c)$.
Example: Addition of integers is associative: $(1+2)+3=6=1+(2+3)$. Notice however, that subtraction is not associative. Indeed, $2=1-(2-3)\neq (1-2)-3=-4$.
Definition: A binary operation $*$ on $A$ is commutative is for all $a,b\in A$, $a*b=b*a$.
Example: Multiplication of rational numbers is commutative: ${\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}={\frac {ca}{bd}}={\frac {c}{d}}\cdot {\frac {a}{b}}$. Notice that division is not commutative: $2\div 3={\frac {2}{3}}$ while $3\div 2={\frac {3}{2}}$. Notice also that commutativity of multiplication depends on the fact that multiplication of integers is commutative as well.

## Exercise

* Of the four arithmetic operations, addition, subtraction, multiplication, and division, which are associative? commutative and identity?

### Answer

| operation | associative | commutative |
| Addition | yes | yes |
| Multiplication | yes | yes |
| Subtraction | No | No |
| Division | No | No |

## Algebraic structures

Binary operations are the working parts of algebraic structures:
Augustus De Morgan expressed the venture into the categories of algebraic structures as follows: It is "inventing a distinct system of unit-symbols, and investigating or assigning relations which define their mode of action on each other." De Morgan 1844, quoted by A.N. Whitehead, A Treatise on Universal Algebra, 1898, page 131.

### One binary operation

A closed binary operation o on a set A is called a magma (A, o ).
If the binary operation respects the associative law a o (b o c) = (a o b) o c, then the magma (A, o ) is a semigroup.
If a magma has an element e satisfying e o x = x = x o e for every x in it, then it is a unital magma. The element e is called the identity with respect to o. If a unital magma has elements x and y such that x o y = e, then x and y are inverses with respect to each other.
A magma for which every equation a x = b has a solution x, and every equation y c = d has a solution y, is a quasigroup. A unital quasigroup is a loop.
A unital semigroup is called a monoid. A monoid for which every element has an inverse is a group. A group for which x o y = y o x for all its elements x and y is called a commutative group. Alternatively, it is called an abelian group.

### Two binary operations

A pair of structures, each with one operation, can used to build those with two: Take (A, o ) as a commutative group with identity e. Let A_ denote A with e removed, and suppose (A_ , * ) is a monoid with binary operation * that distributes over o:
    a * (b o c) = (a * b) o (a * c). Then (A, o, * ) is a ring.
In this construction of rings, when the monoid (A_ , * ) is a group, then (A, o, * ) is a division ring or skew field. And when (A_ , * ) is a commutative group, then (A, o, * ) is a field.
The two operations sup (v) and inf (^) are presumed commutative and associative. In addition, the absorption property requires: a ^ (a v b) = a, and a v (a ^ b) = a. Then (A, v, ^ ) is called a lattice.
In a lattice, the modular identity is (a ^ b) v (x ^ b) = ((a ^ b) v x ) ^ b. A lattice satisfying the modular identity is a modular lattice.

### Three binary operations

A module is a combination of a ring and a commutative group (A, B), together with a binary function A x B → B. When A is a field, then the module is a vector space. In that case A consists of scalars and B of vectors. The binary operation on B is termed addition.

### Four binary operations

Suppose (A, B) is a vector space, and that B has a second operation called multiplication. Then the structure is an algebra over the field A.