Revamp the subsections on preoredered and partially ordered sets

bibliography/books.bib

@@ -46,6 +46,15 @@ @book{Birkhoff1948
   year = {1948}
 }
 
+@book{Birkhoff1967,
+  author = {Garrett Birkhoff},
+  isbn = {978-0-8218-1025-5},
+  language = {english},
+  publisher = {American Mathematical Society},
+  title = {Lattice Theory},
+  year = {1967}
+}
+
 @book{Bollobas1998,
   author = {Béla Bollobás},
   isbn = {978-0-387-98488-9},
@@ -76,6 +85,16 @@ @book{Clarke2013
   year = {2013}
 }
 
+@book{DaveyPriestley2002,
+  author = {Brian Davey and Hilary Priestley},
+  edition = {2},
+  isbn = {978-0-521-78451-1},
+  language = {english},
+  publisher = {Cambridge University Press},
+  title = {Introduction to lattices and order},
+  year = {2002}
+}
+
 @book{Deimling1985,
   author = {Klaus Deimling},
   isbn = {3540139281},
@@ -179,6 +198,28 @@ @book{GondranMinoux1984Graphs
   year = {1984}
 }
 
+@book{Gratzer2011,
+  author = {George Grätzer},
+  doi = {10.1007/978-3-0348-0018-1},
+  isbn = {9783034800181},
+  language = {english},
+  publisher = {Springer Basel},
+  series = {[]},
+  title = {Lattice Theory: Foundation},
+  url = {http://dx.doi.org/10.1007/978-3-0348-0018-1},
+  year = {2011}
+}
+
+@book{Harzheim2005,
+  author = {Egbert Harzheim},
+  isbn = {0-387-24219-8},
+  language = {english},
+  publisher = {Springer},
+  series = {Advances in Mathematics},
+  title = {Ordered Sets},
+  year = {2005}
+}
+
 @book{Hinman2005,
   author = {Peter G. Hinman},
   isbn = {1-56881-262-0},
@@ -208,6 +249,15 @@ @book{Jacobson1985Vol2
   year = {1985}
 }
 
+@book{Johnstone1982,
+  author = {Peter T. Johnstone},
+  isbn = {9780521238939},
+  language = {english},
+  publisher = {Cambridge University Press},
+  title = {Stone spaces},
+  year = {1982}
+}
+
 @book{Kaplansky1974Rings,
   author = {Irving Kaplansky},
   isbn = {0226424545},
@@ -507,6 +557,17 @@ @book{ГеновМиховскиМоллов1991
   year = {1991}
 }
 
+@book{Гуров2013,
+  author = {Сергей Исаевич Гуров},
+  bbc = {22.144},
+  language = {russian},
+  shortauthor = {Sergey Isаevich Gurov},
+  subtitle = {Определения, свойства, примеры},
+  title = {Булевы алгебры, упорядоченные множества, решетки},
+  udc = {512(075.8)},
+  year = {2013}
+}
+
 @book{Гюзелев1873,
   author = {Иван Гюзелев},
   language = {bulgarian},
@@ -526,6 +587,30 @@ @book{ДочевДимитровЧуканов1976
   year = {1976}
 }
 
+@book{Зорич2019Том1,
+  author = {Владимир Антонович Зорич},
+  edition = {10},
+  language = {russian},
+  publisher = {МЦНМО},
+  shortauthor = {Vlаdimir Аntonovich Zorich},
+  title = {Математический анализ},
+  url = {https://matan.math.msu.su/media/uploads/2020/03/V.A.Zorich-Kniga-I-10-izdanie-Corr.pdf},
+  volume = {1},
+  year = {2019}
+}
+
+@book{Зорич2019Том2,
+  author = {Владимир Антонович Зорич},
+  edition = {9},
+  language = {russian},
+  publisher = {МЦНМО},
+  shortauthor = {Vlаdimir Аntonovich Zorich},
+  title = {Математический анализ},
+  url = {https://matan.math.msu.su/media/uploads/2020/03/V.A.Zorich-Kniga-II-9-izdanie-Temp-Corr-3.pdf},
+  volume = {2},
+  year = {2019}
+}
+
 @booklet{ИвановТужилин2017,
   author = {Александр Олегович Иванов and Алексей Августинович Тужилин},
   language = {russian},
@@ -535,6 +620,18 @@ @booklet{ИвановТужилин2017
   year = {2017}
 }
 
+@book{ИльинСадовничийСендов1985Том1,
+  author = {Владимир Александрович Ильин and Виктор Антонович Садовничий and Благовест Христов Сендов},
+  edition = {2},
+  language = {russian},
+  publisher = {Издательство Московского университета},
+  shortauthor = {Vlаdimir Аleksаndrovich Ilin and Viktor Аntonovich Sаdovnichiy and Blаgovest Khristov Sendov},
+  subtitle = {Начальный курс},
+  title = {Математический анализ},
+  volume = {1},
+  year = {1985}
+}
+
 @book{ИоффеТихомиров1974,
   author = {Александр Давидович Иоффе and Владимир Михайлович Тихомиров},
   language = {russian},
@@ -674,6 +771,17 @@ @book{ПетровЗяпков2010
   year = {2010}
 }
 
+@book{Проданов1982,
+  author = {Иван Проданов},
+  language = {bulgarian},
+  publisher = {Наука и изкуство},
+  shortauthor = {Ivаn Prodаnov},
+  subtitle = {Построение на някои класически функционални пространства},
+  title = {Увод във функционалния анализ},
+  volume = {1},
+  year = {1982}
+}
+
 @book{Тагамлицки1971Диф,
   author = {Ярослав Тагамлицки},
   edition = {5},
@@ -685,6 +793,17 @@ @book{Тагамлицки1971Диф
   year = {1971}
 }
 
+@book{Тагамлицки1978Инт,
+  author = {Ярослав Тагамлицки},
+  edition = {6},
+  language = {bulgarian},
+  publisher = {Наука и изкуство},
+  shortauthor = {Yaroslаv Tаgаmlitski},
+  title = {Интегрално смятане},
+  url = {https://store.fmi.uni-sofia.bg/fmi/or/_p2.pdf},
+  year = {1978}
+}
+
 @book{Тыртышников2007,
   author = {Евгений Тыртышников},
   isbn = {978-5-9221-0778-5},

bibliography/papers.bib

@@ -14,6 +14,18 @@ @article{Anderson1988
   year = {1988}
 }
 
+@misc{AhoGareyUllman1972,
+  author = {Alfred V. Aho and Michael R. Garey and Jeffrey D. Ullman},
+  doi = {10.1137/0201008},
+  language = {english},
+  month = {6},
+  primaryclass = {General Mathematics},
+  publisher = {Society for Industrial & Applied Mathematics (SIAM)},
+  title = {The Transitive Reduction of a Directed Graph},
+  url = {http://dx.doi.org/10.1137/0201008},
+  year = {1972}
+}
+
 @article{BezhanishviliHolliday2019,
   author = {Guram Bezhanishvili and Wesley H. Holliday},
   doi = {10.1016/j.indag.2019.01.001},

figures/ex__common_polynomial_divisors__incomparable.tex

@@ -4,7 +4,7 @@
   \begin{tikzcd}[every arrow/.append style={dash}]
         &     &                                    & X^6 &               \\
         & X^5 &                                    &     &               \\
-        &     &                                    &     & X^4 \ar[luu]  \\
+        &     &                                    &     & X^4 \ar[luu] \\
         &     & X^3 \ar[luu] \ar[ruuu]             &     &               \\
         &     & X^2 \ar[luuu] \ar[rruu] \ar[ruuuu] &     &
   \end{tikzcd}

figures/ex__def__partial_order_chain__binary_power_set.tex

@@ -0,0 +1,9 @@
+\documentclass{classes/tikzcd}
+
+\begin{document}
+  \begin{tikzcd}[every arrow/.append style={dash}]
+              & \set{ a, b }                &              \\
+    a \ar[ur] &                             & b \ar[ul] \\
+              & \varnothing \ar[ul] \ar[ur] &
+  \end{tikzcd}
+\end{document}

figures/ex__def__partial_order_chain__ternary_power_set.tex

@@ -0,0 +1,10 @@
+\documentclass{classes/tikzcd}
+
+\begin{document}
+  \begin{tikzcd}[every arrow/.append style={dash}]
+                             & \set{ a, b, c }                    &                          \\
+    \set{ a, b } \ar[ur]     & \set{ a, c } \ar[u]                & \set{ b, c } \ar[ul]     \\
+    \set{ a } \ar[u] \ar[ur] & \set{ b } \ar[ul] \ar[ur]          & \set{ c } \ar[ul] \ar[u] \\
+                             & \varnothing \ar[u] \ar[ul] \ar[ur] &
+  \end{tikzcd}
+\end{document}

text/affine_spaces.tex

@@ -185,7 +185,7 @@ \subsection{Affine spaces}\label{subsec:affine_spaces}
 
   This is an affine combination of members of \( S \). Therefore, \( A(A(S)) = A(S) \).
 
-  \SubProofOf[def:order_homomorphism/increasing]{monotonicity} If \( S_1 \subseteq S_2 \), then \( H(S_2) \) contains the affine combinations of the members of \( S_1 \) in addition to others. Hence, \( H(S_1) \subseteq H(S_2) \).
+  \SubProofOf[def:order_function/preserving]{monotonicity} If \( S_1 \subseteq S_2 \), then \( H(S_2) \) contains the affine combinations of the members of \( S_1 \) in addition to others. Hence, \( H(S_1) \subseteq H(S_2) \).
 \end{defproof}
 
 \begin{definition}\label{def:affine_subspace}\mcite[25]{Gallier2011}

text/banach_space_interpolation.tex

@@ -315,7 +315,7 @@ \subsection{Banach space interpolation}\label{subsec:banach_space_interpolation}
   The \hyperref[def:k_functional]{\( K \)-functional} has the following basic properties:
 
   \begin{thmenum}
-    \thmitem{def:k_functional_properties/basic} For any fixed \( x \in \Sigma \overline{X} \), the function \( t \mapsto K(t, x) \) is positive, \hyperref[def:order_homomorphism]{monotone} and \hyperref[def:convex_functions]{concave}.
+    \thmitem{def:k_functional_properties/basic} For any fixed \( x \in \Sigma \overline{X} \), the function \( t \mapsto K(t, x) \) is positive, \hyperref[def:order_function]{monotone} and \hyperref[def:convex_functions]{concave}.
 
     \thmitem{def:k_functional_properties/inequality} For positive real numbers \( t, s > 0 \), we have the following inequality:
     \begin{equation}\label{eq:def:k_functional_properties/inequality}

text/boolean_algebras.tex

@@ -1,7 +1,7 @@
 \subsection{Boolean algebras}\label{subsec:boolean_algebras}
 
-\begin{definition}\label{def:heyting_algebra}\mcite[147]{Birkhoff1948}
-  A \term{Heyting algebra} or \term{relatively pseudo-complemented algebra} is a \hyperref[def:semilattice/bounded]{bounded} \hyperref[def:semilattice/distributive_lattice]{distributive lattice} \( L \) with a binary operation \( \rightarrow \) defined as
+\begin{definition}\label{def:heyting_algebra}\mcite[8]{Johnstone1982}
+  A \term{Heyting algebra} is a \hyperref[def:semilattice/lattice]{lattice} \( L \) with a binary operation \( \rightarrow \) defined as
   \begin{equation}\label{eq:def:heyting_algebra/conditional}
     (x \rightarrow y) \coloneqq \bigvee\set{ a \in L \given a \wedge x \leq y }.
   \end{equation}
@@ -23,9 +23,9 @@ \subsection{Boolean algebras}\label{subsec:boolean_algebras}
 
   Heyting algebras have the following metamathematical properties:
   \begin{thmenum}[resume=def:heyting_algebra]
-    \thmitem{def:heyting_algebra/theory} We extend the language of the \hyperref[def:semilattice/theory]{theory of lattices} with the binary infix functional symbol \( \rightarrow \) and the unary functional symbol \( \widetilde{\anon} \). By adding the axiom \eqref{eq:def:heyting_algebra/conditional} to the theory of bounded distributive lattices, we obtain the theory of Heyting algebras.
+    \thmitem{def:heyting_algebra/theory} We extend the language of the \hyperref[def:semilattice/theory]{theory of lattices} with the binary infix functional symbol \( \rightarrow \) and the unary functional symbol \( \widetilde{\anon} \). By adding the axiom \eqref{eq:def:heyting_algebra/conditional} to the theory of lattices, we obtain the theory of Heyting algebras.
 
-    \thmitem{def:heyting_algebra/submodel} The Heyting subalgebras are the \hyperref[def:semilattice/submodel]{bounded sublattices} for which the conditional is well-defined.
+    \thmitem{def:heyting_algebra/submodel} The Heyting subalgebras are the \hyperref[def:semilattice/submodel]{sublattices} for which the conditional is well-defined.
 
     \thmitem{def:heyting_algebra/homomorphism} \hyperref[def:first_order_homomorphism]{First-order homomorphisms} between Heyting algebras are lattice homomorphisms with the additional requirement that homomorphisms preserve conditionals.
 
@@ -34,6 +34,13 @@ \subsection{Boolean algebras}\label{subsec:boolean_algebras}
     \thmitem{def:heyting_algebra/opposite} The \hyperref[def:semilattice/duality]{principle of duality for lattices} does not hold for Heyting algebras.
   \end{thmenum}
 \end{definition}
+\begin{comments}
+  \item Heyting algebras are called \enquote{Browerian} by \incite[147]{Birkhoff1967}, while \incite[7]{Golan2010} defines a Heyting algebra to be what we call a \hyperref[def:category_of_small_frames]{frame}.
+\end{comments}
+
+\begin{proposition}\label{thm:heyting_algebra_is_distributive}
+  A \hyperref[def:heyting_algebra]{Heyting algebra} is necessarily \hyperref[def:semilattice/distributive_lattice]{distributive}.
+\end{proposition}
 
 \begin{example}\label{ex:topological_space_is_heyting_algebra}
   Somewhat similar to how the power set of a nonempty set is a Boolean algebra, as shown in \fullref{thm:boolean_algebra_of_subsets}, the topology \( \mscrT \) of a \hyperref[def:topological_space]{topological space} \( (X, \mscrT) \) is a Heyting algebra. This is actually used in topological semantics --- see \fullref{def:propositional_topological_semantics}.

text/category_equivalences.tex

@@ -479,7 +479,7 @@ \subsection{Category equivalences}\label{subsec:category_equivalences}
     \end{aligned}
   \end{equation*}
 
-  \ref{def:functor/CF1} is immediate and \ref{def:functor/CF2} follows from \eqref{eq:def:order_homomorphism/increasing}, hence \( F \) is indeed a functor.
+  \ref{def:functor/CF1} is immediate and \ref{def:functor/CF2} follows from \eqref{eq:def:order_function/preserving}, hence \( F \) is indeed a functor.
 
   \SubProof{Proof that preorder categories induce preordered sets} Let \( \cat{P} \) be a small category. Define the binary relation
   \begin{equation*}

text/complete_ordered_sets.tex

@@ -36,7 +36,7 @@ \subsection{Complete ordered sets}\label{subsec:complete_ordered_set}
 
   Therefore, \( A^U = A^{ULU} \) and \( A^{UL} = A^{ULUL} \).
 
-  \SubProofOf[def:order_homomorphism/increasing]{monotonicity} Fix subsets \( A \subseteq B \) of \( P \) and a member \( a' \) of \( A^{UL} \). We will show that \( a' \) is also a member of \( B^{UL} \).
+  \SubProofOf[def:order_function/preserving]{monotonicity} Fix subsets \( A \subseteq B \) of \( P \) and a member \( a' \) of \( A^{UL} \). We will show that \( a' \) is also a member of \( B^{UL} \).
 
   First note that \( B^U \subseteq A^U \) because an upper limit of \( B \) is necessarily also an upper limit of \( A \).
 
@@ -112,7 +112,7 @@ \subsection{Complete ordered sets}\label{subsec:complete_ordered_set}
   \end{equation*}
   of \hyperref[def:dedekind_macnielle_closure]{Dedekind-MacNeille-closed sets} ordered via \hyperref[def:subset]{set inclusion}.
 
-  The set \( P \) itself can be embedded via the \hyperref[def:order_homomorphism/increasing]{order-preserving map} which sends each member of \( P \) to its \hyperref[def:order_interval/ray]{initial segment}, that is,
+  The set \( P \) itself can be embedded via the \hyperref[def:order_function/preserving]{order-preserving map} which sends each member of \( P \) to its \hyperref[def:order_interval/unbounded]{initial segment}, that is,
   \begin{equation*}
     \begin{aligned}
       &\iota: P \to D(P), \\
@@ -261,7 +261,7 @@ \subsection{Complete ordered sets}\label{subsec:complete_ordered_set}
 \begin{theorem}[Dedekind-MacNeille completion universal property]\label{thm:dedekind_macneille_completion_universal_property}\mcite[thm. 12]{Birkhoff1948}
   The \hyperref[def:dedekind_macnielle_completion]{Dedekind-MacNeille completion} \( D(P) \) of a \hyperref[def:partially_ordered_set]{partially ordered set} \( P \) satisfies the following \hyperref[rem:universal_mapping_property]{universal mapping property}:
   \begin{displayquote}
-    For every \hyperref[def:semilattice/lattice]{complete lattice} \( L \), every \hyperref[def:order_homomorphism/increasing]{order-preserving map} \( \varphi: P \to L \) \hyperref[def:factors_through]{uniquely factors through} \( D(P) \). More precisely, there exists a unique \hyperref[def:semilattice/homomorphism]{lattice homomorphism} \( \widetilde{\varphi}: D(P) \to L \) such that the following diagram commutes:
+    For every \hyperref[def:semilattice/lattice]{complete lattice} \( L \), every \hyperref[def:order_function/preserving]{order-preserving map} \( \varphi: P \to L \) \hyperref[def:factors_through]{uniquely factors through} \( D(P) \). More precisely, there exists a unique \hyperref[def:semilattice/homomorphism]{lattice homomorphism} \( \widetilde{\varphi}: D(P) \to L \) such that the following diagram commutes:
     \begin{equation}\label{eq:thm:dedekind_macneille_completion_universal_property/diagram}
       \begin{aligned}
         \includegraphics[page=1]{output/thm__dedekind_macneille_completion_universal_property}
@@ -290,7 +290,7 @@ \subsection{Complete ordered sets}\label{subsec:complete_ordered_set}
     \widetilde{\varphi}(P_{\leq x})
     =
     \bigvee\nolimits^L \set{ \varphi(a) \given a \in P_{\leq x} }
-    \reloset {\eqref{eq:def:order_homomorphism/increasing}} =
+    \reloset {\eqref{eq:def:order_function/preserving}} =
     \varphi(x).
   \end{equation*}
 
@@ -401,7 +401,7 @@ \subsection{Complete ordered sets}\label{subsec:complete_ordered_set}
 \end{remark}
 
 \begin{theorem}[Dedekind completion]\label{thm:def:dedekind_completion}
-  The \hyperref[def:dedekind_completion]{Dedekind completion} of an \hyperref[def:extremal_points/upper_and_lower_bounds]{unbounded} \hyperref[def:totally_ordered_set]{totally ordered set} is, up to an \hyperref[def:order_homomorphism/isomorphism]{order isomorphism}, the smallest such set that is \hyperref[def:dedekind_completeness]{Dedekind complete}.
+  The \hyperref[def:dedekind_completion]{Dedekind completion} of an \hyperref[def:extremal_points/upper_and_lower_bounds]{unbounded} \hyperref[def:totally_ordered_set]{totally ordered set} is, up to an \hyperref[def:order_function/isomorphism]{order isomorphism}, the smallest such set that is \hyperref[def:dedekind_completeness]{Dedekind complete}.
 \end{theorem}
 \begin{proof}
   \SubProof{Proof of completeness} Let \( (\mscrA, \mscrB) \) be a \hyperref[def:dedekind_cut]{cut} in the Dedekind completion. We will show that \( \mscrA \) has a maximum or \( \mscrB \) has a minimum.