# Mathematics Prelims

## January 18, 2009

### Introductory Algebra and Definitions

Filed under: Abstract Algebra,Algebra — cjohnson @ 10:21 am

If $S$ is a set, a binary operation on $S$ is a function $\circ : S \times S \to S$, where we generally write $s \circ t$ for $\circ(s, t)$.  We say that this binary operation is associative if for every $s,t,u \in S$ we have that $(s \circ t) \circ u = s \circ (t \circ u)$.  A pair $(S, \circ)$ where $S$ is a set and $\circ$ an associative binary operation on $S$ is then called a semigroup.  For example, $(\mathbb{N}, +)$ is a semigroup.

If $(S, \circ)$ is a semigroup and there exists an $e \in S$ such that for every $s \in S$ we have that $s \circ e = e \circ s = s$, then we say that $e$ is an identity (of $\circ$) and we call $(S, \circ)$ a monoid.  The natural numbers together with zero, which we’ll denote $\mathbb{N}_0$, is then a monoid under addition.

Generally we’ll let juxtaposition denote the binary operation, writing $st$ for $s \circ t$.  In cases where $+$ is the “natural” symbol to use for the binary operation, however, (e.g., the natural numbers with addition), then we’ll still use $+$.

When $s,t \in S$ and $st = e$ (where $e$ still denotes the identity), then we say that $s$ is the inverse of $t$ (and vice versa).  We will normally write $s^{-1}$ to mean the inverse of $s$, though when $+$ is the operation we’ll write $-s$ and write $t - s$ to mean $t + (-s)$.  A monoid where every element has an inverse is called a group.  The integers under addition form the group $(\mathbb{Z}, +)$.  To check this, simply note that addition of integers is associative, addition of two integers yields another integer, zero is the identity element, and for any $z \in \mathbb{Z}$ we have that $-z$ is its inverse.