\part{Advanced}


\chapter{Lattices}

Most of

\section{Poset representation of a lattice}

\emph{join} is the folloiwng operation (when its result exists).
\[p \lor q = \sup (\{p,q\})\]

\emph{meet} is the following operation (when its result exists).
\[p \land q = \inf (\{p,q\})\]


Recall that although for any subset of a poset its supremums and infimums are unique (if it exists), existence may not hold. This is because there may be o upper (lower) bounds to begin with, or if they do, there may not be a least (greatest) element among those bounds. If we consider posets where joins or meets are guaranteed to exist, we introduce some rather nice structures.


A \emph{join-semilattice} is a poset on which all joins are defined.
A \emph{meet-semilattice} is a poset on which all meetsare defined.


\begin{definition}[(Poset) lattice]
A \emph{(poset) lattice} is a poset that is both a join and meet semilattice.
\end{definition}


$[a \leq b ]\implies [a \land c \leq b \land c]$
$[a \leq b ]\implies [a \land c \leq b \lor c]$
$[a \leq b ] \land [c \leq d] \implies [a \lor c \leq b \lor d]$
$[a \leq b ] \land [c \leq d] \implies [a \land c \leq b \land d]$

Let $(T,\leq)$ be a toset, then it is a lattice.

\subsection{Number theory example}

Take the poset $(\mathbb{N},|)$ (where everything divides 0), we see this is a lattice since $a \lor b = \gcd (a,b), a \land b = \lcm (a,b)$ are always defined.





\section{Algebraic representation}


Since the join and meet can be seen as operations well defined on the poset, this inspires us to seek what 'algebraic laws' the join and meet obey.
We will eventually .





\begin{definition}[(Algebraic) lattice]
A \emph{(algebraic) lattice} is an algebraic structure $(L,\land, \lor)$
$a \land b = b \land a$
$a \lor b = b \lor a$
$a \land (b \land c) = (a \land b) \land c$
$a \lor (b \lor c) = (a \lor b) \lor c$

$a \land (a \lor b) = a$
$a \lor (a \land b) = a$
\end{definition}



a poset lattice gives rise to an algebraic lattice and an algebraic lattice gives rise to an algebraic lattice. Since they are two interpretations of the same structure, one often dropps the 'poset' or 'algebraic' before the 'lattice'.



We can see some elementary conseqeunces of a lattice.

\begin{proposition}
$a \land a = a$
$a \lor a = a$
\end{proposition}



\section{Complete lattice}


We have been viewing joins and meets as binary operations, but like $+ \to \sum$,$\times \to \prod$, we can take the join of arbitrary elements over a set. A lattice



\begin{definition}[Complete lattice]
The join of any set exists
The meet of any set exists
\end{definition}

In real analysis one encounters Dedekind completeness, which is very similar to he definition of a complete lattice although requires joins and meets to exist only when such sets have lower and upper bounds to begin with.

\begin{definition}[Dedekind complete lattice]
The join of any set with lower bounds exists
The meet of any set with upper bounds exists
\end{definition}


\section{Sublattices}
\section{Lattice homomorphisms}


\begin{definition}[Lattice homomorphism]
\end{definition}

\begin{corollary}
Lattice homomorphisms are monotone.
\end{corollary}

\section{Lattice ideals}
\section{Lattice filters}
\section{Knaster-Tarski theorem}

Let $(L,\leq)$ be a complete lattice with monotone function $f$, then the fixed points of $f$ form a complete lattice under $\leq$



\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