\chapter{Operations}

\begin{definition}
An \emph{operation} is a function with its domain as its image; it maps a set to itself. it may take input values called \emph{operands}.
\end{definition}
- nullary operation has 0 operands
- unary operation 1 operand
- binary operation has 2 operands
- n-ary operation has $n$ operands
- equational law

- associative operation
\[ \cdot \text{ is associative } \iff x \cdot (y \cdot z) = (x \cdot y) \cdot z \]
- commutative operation
\[ \cdot \text{ is commutative } \iff x \cdot y = y \cdot x\]
- left distributive operation
\[ \cdot \text{ is left distributive with } + \iff x \cdot ( y + z ) =  ( x \cdot  y ) + ( x \cdot  z ) \]
- right distributive operation
\[ \cdot \text{ is right distributive with } + \iff ( y + z ) \cdot x =  ( y \cdot x ) + ( z \cdot  x ) \]




Here are some common existential laws regarding elements in the domain of binary operations.

\begin{definition}
An element is an \emph{identity element} iff when paired with any element returns said element.
\[1 \text{ is an identity element of } \cdot \iff \forall x ( 1 \cdot x = x \cdot 1 = x ) \]
\end{definition}
\begin{definition}
\emph{invertible element} iff there is some other element that when paired with the operation returns the identity element.
\[x \text{ is invertible in } \cdot \iff \exists x^{-1} ( x^{-1} \cdot x = x \cdot x^{-1} = 1)\]
\end{definition}
