\chapter{Operations}

When learning abstract algebra, there are a myriad of different structures one can study, each forming their own field (group theory, ring theory, linear algebra etc.) and subfields may be based on certain techniques (Galois theory).

Universal algebra aims to study the general construction and behaviour of algebraic structures in general, as opposed to specific ones such as groups or rings.

Algebraic structures will be defined as a tuple of a set and operators, so before we can begin this branch of mathematics we require a strong understanding of operators (basic set theory is assumed).

\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

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