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\begin{document}
\title{Notes}
\date{January 31, 2014}
\maketitle
\section*{4.5}
\subsection*{what is a partially ordered set (poset)?}

a poset is a special kind of relation.
\subsubsection*{def}
A relation $R$ on a set $X$ is a subset of $X\cross X$. We say $xRy$ if $(x,y)\in R$"x is related to y"
\subsubsection*{properties}
reflexive: $xRx$

symmetric: $xRy\to yRx$

antisymetric: $xRy\to y\slash{R}x, x\neq y$

transitive: $xRy\to yRz\to xRz$

equivalence relations: reflexive, symmetric, transitive.

a poset is a relation that is reflexive, antisymmetric, transitive.

a poset is a set with a partial order "$\leq$" that is reflexive, antisymmetric and transitive.

explain it like i'm 7: rule is graph has to have arrows going in the same way, no cycles.

\subsubsection*{example}
set is positive integers. partial order is divisibility.
$a\leq b$ if $a\|b$

$2\leq 6$ but $5\slash\leq 6$. 5 and 6 are incomparable. prove it is a poset

$l,m,n\in \mathbb{N}$ then $n\|n$, so reflexive. suppose $m\|n$ and $m\neq n$ then $\backE a\in\mathbb{N}$ such that $am=n$ so $m<n$. so $n\slash \| m$. if $l\|m$ and $m\|n$ then blah blah $l\|n$
\subsubsection*{example}
subsets of $\{a,b,c\}$ with partial order of set containment

definition: in a poset we say y covers x if $x<y$ and if $x\leq z\leq y$ then $x=z$ or $z=y$.

definition: the hassediagram of a poset is the (directed) graph of the covers


\subsection*{weak bruhat order}
set is permutations of $[n]$. sometimes easier to find covers and extrapolate order.

covers: y covers x in weak bruhat order if $s_i(x)=y$ and the number of inversions of $y$ is one greater than the number of inversions of x.

can hink of permutations of $[n]$ as words in the alphabet $s_1,s_1,\dots,s_{n-1}$. subject to conditions that $s_i^2=1,s_is_j=s_js_i \|i-j\|>1, s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$

this is defining a group, specifically "a presentation of the symmetric group"

to find weak bruhat set for n=4 google. looks like a soccerball or buckyball, "weak order ?"
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