*************************
Split out inessential things:
have a *separate* text on lattices
(and logic etc.)
*************************

key point:
think of sups and infs in terms of lattices
other names (lub and glb) are analogous to lcm and gcd!

\chapter{Sup \& Inf}

...with lots of lattice theory

**********************
Mention the connection between lattices and logic:
$a \leq b$ is $b \implies a$,
and $\vee, \wedge$ are $\and$ and $\or$

Thus different kinds of lattices (and generalizations)
yield different kinds of logic,
...and have geometric models!
[e.g.:
- discrete topology = classical logic
- topology = constructible logic
]
**********************

\section{Definition}
Given a poset $(P,\leq)$, the \def{supremum} (or $\sup$) of a subset $A \subset P$
is its least upper bound, if this exists.
That is, $a \leq s$ for all $a \in A$ (upper bound)
and $s$ is minimal among these: $s \leq b$ for any upper bound $b$. 

(for any poset)
least upper bound
upper bound,
least among those

inf is the opposite notion (or, to be clever, the same notion on the opposite order)
greatest lower bound

{ped}
. very formal
. dry examples
(it's *dry*: it's very Rudin-esque [pun on Rubenesque intended])

\emph{Yes,} abstract reasoning is a key part of math
(discrete math, esp. basic number theory and algebra,
 is the best place to teach this)
Conflict between *material* and *method/way-of-thought*:
calc and linear algebra are taught b/c the *material* is very useful,
but they are a poor place to learn the *method* of math.
{ped}


\section{Examples}
finite infs/sups on linear set = min/max
on product lattice: gcd/lcm (very graphical/2D), set intersection/union
[one aside for gcd/lcm:
 for these, ``least'' can be w/r/t the divisibility order (which is more correct),
 or w/r/t the Archimedean order -- doesn't matter, but possibly confusing.]

\begin{ped}
Formal properties much clearer with lcm & gcd b/c *finite*,
and several dimensions: hard to see things in calculus (basic analysis)
b/c all infinite and 1-dimensional.
\end{ped}
[also issue is "not enough examples":
 egs for subsets of rationals/reals are boring]

\begin{ped}
*draw* some posets
...and have students identify sups, infs, greatest elements, maximal elements,
etc., etc.
\end{ped}


Any poset that's not bounded above doesn't have arbitrary sups and infs
e.g., \bR, finite subsets of an infinite set, integers under divisibility
[``completing'' \bR with +\infty]

\section{Topological Interpretation}
For $\bR$, it's just max of closure
(\& inf is min of closure)
The definition is: "min of upper bounds":
it's basically min of a complement,
and it's a rather round-about way.

\section{Background: lattices}
[notionally best is:
 a \geq b iff b \leq a;
 using \leq' is formally correct and completely confusing]

[note on natural poset structure on product of posets;
 lex order is a natural total order on product of total orders]
...but not on Q (but on R, yes: this is completion)

NB: order (like equiv) is best a *closed* notion (reflexive);
e.g. of 2|-2 and -2|2

affinely extended real number system \bar \bR
with +/- infty
``ideal points'', which are *not* real numbers
affine closure, 2-point compactification
(there's also
projectively extended real number system; \hat \bR
)
partially extend arithmetic

[EG of product order;
 look at circle in product order,
 and find upper bounds, least upper bound,
 maximal elements, but no greatest element]

meet/join are algebraic abstractions of intersection/union
(hence have angular symbols instead of curved: \subset vs. < ;-)

[these operations are symmetric and associative
 for sets b/c boolean logic is!]
notational note: \subseteq is analogous to \leq
  \subset generally means \subseteq (we write as latter to make analogy explicit)
  ...though notation would suggest \subset = <
  (instead, it's written as \subsetneq)

poset -> lattice;
lattice -> poset (a \leq b iff a \join b = b iff a \meet b = a;
  EGs:
. sets: a \subseteq b iff a \cap b = a iff a \cup b = b
. linear order: a \leq b iff \min(a,b) = a, iff \max(a,b)=b
. divisibility: a|b iff gcd(a,b)=a iff lcm(a,b)=b
)

NB: \emph{the} greatest element means \forall s, s \leq g
  (necessarily unique)
\emph{a} maximal element means \nexists s s.t. m < s
  can have lots of maximal elements
  distinct maximal elements are incomparable
(for total order they coincide, since always comparable)

The point is:
- a greatest element does *not* always exist,
  even if a sup does,
  b/c sup may not be element of set
(urk: talking about minimal upper bounds vs. *least* upper bound)
[better to talk about greatest/least,
 and then maximal/minimal]
point is:
- bounded (above) subsets of integers do *not* always have a greatest element
- ...but their set of upper bounds *does* have a *least* element

BTW, a *topology* (axiomatized in terms of open sets)
is simply a special kind of sublattice of $\cP(X)$.
Abstracting yields a locale:
bottom, top, finite meets, arbitrary joins

\section{Applications}
. measure (inner & outer measure)
. Dirichlet POV on integrals (lower/upper integral)
{ped}
BTW, this is formally very back-and-forth: lower integral
is min of upper bounds of lower functions
Intuitively very simple though: at least as big as all lower functions

NB on this being a *closed* thing:
  closed is about *limits*

. uniform metric
  (or basically anywhere you want to use a max)
. continuity/oscillation of a function or convergence of sequence (lim
inf/lim sup)
[cool: a sequence converges iff oscillation = 0; lim inf = lim sup (and
both exist);
 a function is continuous at x iff osc = 0;
 recall that osc is good b/c rather than having 2 quantifiers,
 you've packed it into a concept and a limit]
[a sequence is constant iff min = max (& both exist; NB: also define/say
bounded)]

(optimization? general analysis trick of bounding?)


completeness -> topology of reals:
IVT
maximum principle
(e.g., 2-x^2)
boundedness does *not* require these

\section{Motivations}
Key EG of rigor in analysis (19th century, b/c people made mistakes)
. Weierstrass and calc of variations (function realizing inf may not
exist)
. IVT (rigorously)
..> Dedekind cuts (most simply, just complete w/r/t order;
Dedekind cuts are a concrete model / construction
{ped}
That is, we define/think about reals abstractly, axiomatically;
need to prove that it works / that it's coherent.
[For items with physical models, this is generally glossed over:
 of course the reals work, as we have real lengths, etc.;
 this is why physicists find rigor less necessary]
{ped}
(better if give EG of incoherent theories,
 and rigorously show coherence of non-obvious theories)

\section{Variants}
$\limsup, \liminf, \esssup$

lim sup = limit superior = limit of supremum
lim inf = limit inferior = limit of infimum

BTW, \limsup and \liminf of sets shows up;
it's a useful notation
$\limsup A_i := \bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i$
(b/c sup for sets is union)
...and dually for \liminf

hmmm...
$\bigcup_{m=1}^\infty \bigcap_{n=m}^\infty \bigcup_{i=n}^\infty A_i$
doesn't give anything new;
want to say that it's like adjointness/complements/perps
(FGF = F and all that: not quite inverses)

\section{My Notes}
This arose when I learned the connection with gcd and lcm.
...and b/c it's pretty basic but pretty confusing.

---------------------------------------------
Problem of the single example
(this is a real issue in Lie theory)
---------------------------------------------

[can you complete a poset? can you complete an ordered field?
 NB: p-adics and F_q are not *ordered*, hence can't do this,
 but they do admit a metric, hence Cauchy/completion POV works;
 the order POV does work and clarify R, but not others.
 BTW, positive is all you need for order on a field;
 distance between two points is the positive difference.
 Since no positive on positive characteristic (b/c have a loop!), no
order:
 positive char is topologically rather *circular*!
 Hey, that's much clearer than previous proofs I've seen;
 for positive char, there's an additive circle; for C, there's a
multiplicative circle.]
