# Abstract Algebra/Binary Operations

A **binary operation** on a set is a function . For , we usually write as .

## Properties[edit | edit source]

The property that for all 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. is not an integer, so the integers are not closed under division.

To indicate that a set has a binary operation defined on it, we can compactly write . 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 is *associative* if for all , .

**Example:** Addition of integers is associative: . Notice however, that subtraction is not associative. Indeed, .

**Definition:** A binary operation on is *commutative* is for all , .

**Example:** Multiplication of rational numbers is commutative: . Notice that division is not commutative: while . Notice also that commutativity of multiplication depends on the fact that multiplication of integers is commutative as well.

## Exercise[edit | edit source]

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

### Answer[edit | edit source]

operation | associative | commutative |
---|---|---|

Addition | yes | yes |

Multiplication | yes | yes |

Subtraction | No | No |

Division | No | No |

## Algebraic structures[edit | edit source]

Binary operations are the working parts of algebraic structures:

### One binary operation[edit | edit source]

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[edit | edit source]

A pair of structures 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**.