my forest
add compactness notes
+109
-2
+6
trees/dm/dm-000R.tree
+6
trees/dm/dm-000R.tree
+3
-2
trees/dt/dt-001Y.tree
+3
-2
trees/dt/dt-001Y.tree
···
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\transclude{dt-001Z}
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\subtree{
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\taxon{Lecture}
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\title{Scott Domains}
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\title{Recursively Defined Domains}
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\p{todo}
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}
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\subtree{
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\taxon{Lecture}
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-
\title{Recursively Defined Domains}
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\title{Scott Domains}
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\transclude{dt-0040}
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\p{todo}
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}
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\subtree{
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trees/dt/dt-003U.tree
+12
trees/dt/dt-003U.tree
···
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\import{dt-macros}
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\title{Compactness}
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\taxon{Definition}
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\author{liamoc}
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\p{
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Let #{A} be a [cpo](dt-001D). Then #{x \in A} is \em{compact} (a.k.a. \em{finite}) iff for all [directed](dt-0010) #{X \subseteq A}:
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##{
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x \sqsubseteq \bigsqcup X \implies \exists y \in X.\ x \sqsubseteq y
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}
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In English: A compact element will approximate some element of a [directed](dt-0010) set if it approximates the [lub](dt-0017). We write #{\compact{A}} for the set of compact elements of #{A}, i.e.:
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##{\compact{A} = \Set{x \in A \mid x\ \text{is compact} }}
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}
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trees/dt/dt-003V.tree
+10
trees/dt/dt-003V.tree
···
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\import{dt-macros}
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\author{liamoc}
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\taxon{Example}
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\title{Compact elements}
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\ul{
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\li{Every element in a finite [cpo](dt-001D) is [compact](dt-003U). More generally, every element of a [cpo](dt-001D) of finite \em{height} (e.g. #{\mathbb{Z}_\bot}) is [compact](dt-003U). This is because finite [directed sets](dt-0010) are always [boring](dt-003W).}
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\li{ Take the [cpo](dt-001D) #{\mathcal{P}(X)} of subsets of #{X} ordered by inclusion #{\subseteq}, as seen in \ref{dt-001G}. The [compact](dt-003U) elements of #{\mathcal{P}(X)} are those of finite cardinality. }
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\li{ Take the [cpo](dt-001D) #{\mathbb{N} \nrightarrow \mathbb{N}} of partial functions on the natural numbers, ordered by inclusion of graphs. The [compact](dt-003U) elements of #{\mathbb{N} \nrightarrow \mathbb{N}} are the functions which are defined only for finite domains. }
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}
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+12
trees/dt/dt-003W.tree
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trees/dt/dt-003W.tree
···
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\import{dt-macros}
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\import{table-macros}
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\author{liamoc}
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\taxon{Definition}
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\title{Boring and interesting sets}
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\p{
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There are two kinds of [directed sets](dt-0010):
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\ol{
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\li{ \em{Boring} sets contain their [lub](dt-0017). \br \strong{Example}: Finite directed sets are boring (but boring sets aren't all finite!). }
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\li{ \em{Interesting} sets don't contain their [lub](dt-0017). \br \strong{Example}: #{\mathbb{N}} in the chain #{\mathbb{N} \cup \Set{ \infty }} is interesting.}
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}
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}
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trees/dt/dt-003X.tree
+17
trees/dt/dt-003X.tree
···
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\import{dt-macros}
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\import{table-macros}
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\title{Compact elements with infinitely many approximations }
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\taxon{Example}
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\author{liamoc}
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\<html:div>[class]{sidefigure}{
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\tex{\usepackage{tikz-cd}\usetikzlibrary{decorations.pathreplacing}\usepackage{amsmath}}{\begin{tikzcd}
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\infty + 1& \ar[l,thick]\text{\scriptsize compact}\\
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\infty\ar[u,-,thick] & \ar[l,thick]\text{\scriptsize non-compact}\\
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2\ar[u,-,thick, dotted] & \quad\arrow[dd, start anchor=north, end anchor=south, no head, xshift=-1em, decorate, decoration={brace,amplitude=10pt,raise=-15pt}, thick,"\text{compact}" right=0pt]\\
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1\ar[u,-,thick]\\
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0\ar[u,-,thick] & \quad
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\end{tikzcd}}
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}
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\p{[Compact](dt-003U) elements may still have an infinite number of approximations. Consider the [cpo](dt-001D) #{\mathbb{N} \cup \Set{ \infty, \infty + 1 }}, where we have tacked on an additional top element #{\infty + 1} to our normal [cpo](dt-001D) of natural numbers extended with infinity. Then, while #{\infty} is not [compact](dt-003U), #{\infty + 1} \em{is} [compact](dt-003U) — it is not the [lub](dt-0017) of an [interesting](dt-003W) [directed set](dt-0010). If #{\infty + 1} is the [lub](dt-0017) of a set #{X} then #{\infty + 1} must be in the set #{X}. Nonetheless, there are an infinite number of \em{approximations} to #{\infty + 1}, i.e., elements #{x} such that #{x \sqsubseteq \infty + 1}.}
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\p{As an aside, requiring that our [compact](dt-003U) elements have a \em{truly} finite number of approximations is the basis for the theory of \em{Berry domains and stable functions}.}
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+4
trees/dt/dt-003Y.tree
+4
trees/dt/dt-003Y.tree
+11
trees/dt/dt-003Z.tree
+11
trees/dt/dt-003Z.tree
···
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\import{dt-macros}
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\author{liamoc}
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\title{Both infinite and compact}
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\taxon{Exercise}
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\p{Find a set in the [cpo of submonoids](dt-003Y) of #{(\mathbb{N},0,+)} that is \em{both} infinite and [compact](dt-003U)}
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\solnblock{
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\p{Take #{E \triangleq \Set{ n \in \mathbb{N} \mid n\ \text{is even}}}. Let #{Y \subseteq \pow{\mathbb{N}}} be directed and #{E \subseteq \bigcup Y}. Since #{2 \in E} there must exist #{y \in Y} such that #{2 \in Y}. Since #{(y,0,+)} is a [monoid](dm-000O), every positive multiple of #{2} is also in #{y}, thus #{E \subseteq y} and therefore #{E} is [compact](dt-003U).}
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\p{Seeing as [compact](dt-003U) elements are also sometimes called \em{finite}, this is surprising as #{E} is an infinite set. The reason it is nonetheless [compact](dt-003U) is that #{E} is \em{finitely generated} — it is the smallest [submonoid](dm-000R) of #{(\mathbb{N},0,+)} such that the \em{finite} set #{\Set{ 2 } \subseteq E}. In fact, the [compact](dt-003U) [submonoids](dm-000R) of #{(\mathbb{N},0,+)} are precisely the finitely generated ones.
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}
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}
+33
trees/dt/dt-0040.tree
+33
trees/dt/dt-0040.tree
···
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\title{Compactness and Finiteness}
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\author{liamoc}
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\p{We begin by formalising the notion of an element in a [cpo](dt-001D) representing a \em{finite} amount of information. }
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\transclude{dt-003W}
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\p{
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Following the above intuition, we might be tempted to say that the \em{"infinite"} elements of a [cpo](dt-001D) are those which are the [lub](dt-0017) of an [interesting](dt-003W) set, but this notion is too weak. Consider this [cpo](dt-001D) #{X}:
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\figure{
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\tex{\usepackage{tikz-cd,amsmath,amssymb}}{
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\begin{tikzcd}
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\infty\\
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2\ar[u,thick,-,dotted]\\
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1\ar[u,thick,-] & & x\ar[uull,thick,-]\\
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0\ar[u,thick,-] \\
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& \bot \ar[ul,thick,-]\ar[uur,thick,-]
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\end{tikzcd}
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}}
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By the above definition, the only infinite element would be #{\infty}, but if we consider the following isomorphic [cpo](dt-001D), ordered by subset inclusion:
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\figure{
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\tex{\usepackage{tikz-cd,amsmath,amssymb}}{
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\begin{tikzcd}
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\mathbb{N}\\
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\{0,1,2\}\ar[u,thick,-,dotted]\\
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\{0,1\}\ar[u,thick,-] & & x\ar[uull,thick,-]\\
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\{0\}\ar[u,thick,-] \\
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& \emptyset \ar[ul,thick,-]\ar[uur,thick,-]
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\end{tikzcd}
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}}
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Then the set #{x} cannot be a finite set, as any finite set would be a subset of one of the finite sets in the chain #{\emptyset \sqsubseteq \Set{ 0 } \sqsubseteq \Set{0,1} \sqsubseteq \cdots}. Thus, it makes more sense for us to call #{x} an \em{"infinite"} element as well.}
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\transclude{dt-003U}
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\p{In the example cpo #{X} above, all the elements except #{\infty} and #{x} would be [compact](dt-003U). Thus, compactness better captures our notion of an element representing a finite amount of information. This understanding of [compact](dt-003U) elements is a generalisation of the notion of a finite element from the theory of algebraic lattices.}
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\transclude{dt-003V}
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\transclude{dt-003X}
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\transclude{dt-003Z}