\part{Lattice theory}
\chapter{Boolean Algebrae}

Propositional logic deals with the connection of propositional variables by logical connectives. One can analyze propositional logic through the perspective of lattice theory by treating truth values as elements and logical connectives as operators on truth values. We seek to establish a general algebra that propositional logic is represented by. 

\begin{definition}\emph{Boolean algebra}
Complemented distributive lattice represented by a 4 tuple of AND OR NOT and a set of elements, of which $\{\top , \bot\}$ is a subset.
\[( B , \land , \lor , \not )\]
AND is associative
OR is associative
AND is commutative
OR is commutative
AND is distributive over OR
OR is distributive over AND
Identitiy element for AND $\top$
Identitiy element for OR $\bot$
Absorption 1
Absorption 2
Complement element for AND $\bot$
Complement element for OR $\top$
\end{definition}

The set of properties we have used to define a Boolean algebra is not minimal; one can equivalently define a Boolean algebra with fewer, yet stronger properties. From our definition of of a Boolean algebra, one can derive further properties.

\begin{theorem}\emph{Further properties of Boolean Algebrae}
Annihilator element for AND
Annihilator element for OR 
AND is idempotent
OR is idempotent
\end{theorem}

\begin{definition}\emph{Trivial boolean algebra}
Boolean algebra where $\top=\bot$
\end{definition}

Propositional logic is concerned with only truth an false, hence it is a Boolean algebra of 2 elements; this is called a 2-element Boolean algebra.

\begin{definition}{2-element Boolean algebra}
Boolean algebra where $B=\{\top,\bot\}$
\end{definition}





\begin{definition}\emph{Boolean ring}
Ring that arises from a Boolean algebra by considering the XOR and AND operators
\[ ( B, \land , \oplus) \]
\end{definition}




\chapter{Heyting Algebrae}
\chapter{Cylindric Algebrae}



lattice theory
- boolean algebra
- heyting algebra

Topology and orders
- banach lattice
- frechet lattice
