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?

Algebraic structures
Binary operations are the working parts of algebraic structures:

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.