\chapter{Algebraic structures}


Recall that operations are functions that operate on the same set as its codomain (inputs and outputs are of the same set). 
The intense study of operations with special properties are the crux of algebra; they are responsible for the methods of elementary algebra as well as group, ring, and lattice theory.

Universal algebra aims to give a high level overview of commom themes that run across all branches of algebra.

In recent times, Category theory has been praised as abstractifying the role of universal algebra by formalizing and analyzing connections between all fields of mathematics


- algebraic structure
An \emph{algebraic structure} is a set $A$  and collection of operations that follow a finite set of properties.

\begin{itemize}
	\item Group
	\item Monoid
	\item Magma
	\item Ring 
	\item Field 
	\item Module
	\item Lattice
\end{itemize}

\chapter{Constructions}
Some of these constructions even insipre constructions within general topology.
\section{Direct product}
Based on the natural construction from the cartesian product of algebraic structures.
$\times$
$\prod_{j \in J}$
\section{Direct sum}
\section{Quotient algebra}



\chapter{Varieties}

A variety or equational class is an algebraic strucutre defined only by equational identities rather than by conditions stated in logic.



Note that even though some algebraic strucutres are classically defined in a certain fashion, there may be an equivalent definition relying soley of equational identities. Such is the case for groups and rings.

\section{Birkoff's HSP theorem}

\section{Isomorphism theorems}