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%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\title{{\bf Lattices, Order}}

\author{Open Mathematics Collaboration\footnote{All {\it authors} with their {\it affiliations} appear at the end of this white paper.}\ \footnote{Corresponding author: \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}} | {\bf Open Mathematics Collaboration}}}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\begin{document}
\maketitle

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\begin{abstract}

\noindent\texttt{LATTICES, ORDER and their underlying definitions and theorems are presented in this white paper (knowledge base).}

\end{abstract}

\begin{center}
\noindent {\footnotesize\textsf{{\bf keywords}: lattices, order, knowledge base}}
\end{center}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\begin{center}{\it\small The most updated version of this white paper is available at}\\\footnotesize{\url{https://osf.io/fzpc4/download}\\\url{https://zenodo.org/record/7267788}}\end{center}

\begin{center}{\texttt Open Mathematics Knowledge Base}\\ {\footnotesize \url{http://omkb.org}}\end{center}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Introduction}

\begin{enumerate}[leftmargin=*,resume]

\item \cite{priestley,gratzer}

\end{enumerate}\newpage

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Notation}

\begin{enumerate}[leftmargin=*,resume]

\item $\Box:=$ necessarily
\item $x\not\leq y:=$ $x\leq y$ is false
\item $\veebar:=$ exclusive or
\item $\exists!:=$ exists exactly one
\item $\exists_{\leq1}:=$ there is at most one
\item\texttt{Def} $:=$ Definition
\item\texttt{Thm} $:=$ Theorem
\item $\cong~:=\,\,$ \texttt{order}-isomorphism
\item $\cong_\ell~:=~$ $\ell${\it attice}-isomorphism
\item $\mathbb N_0=\{0,1,2,...\}$
\item $(A\vdash B):=A$ proves $B$
\item $(A\dashv\vdash B)\equiv(A\vdash B~\land~B\vdash A)$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Order (or partial order)}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(P,\leq):=\hbox{\texttt{order} (or \texttt{partial order}) if }\forall x,y,z\in P$$
\begin{itemize}
\item[($i$)] $x\leq x~$ (reflexive)
\item[($ii$)] $x\leq y~\land~y\leq z~\to~x\leq z~$ (transitive)
\item[($iii$)] $x\leq y~\land~y\leq x~\to~x=y~$ (antisymmetric)
\end{itemize}

\item $P:=$ set;\hspace{0.6cm} $\leq\,\,\,:=$ binary relation on $P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Discrete order}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$=~:=~\texttt{discrete order}$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Quasi-order (pre-order)}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def}\\ \texttt{quasi-order (pre-order)} $:=$ reflexive $\land$ transitive $\land$ $\neg\Box$ antisymmetric

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Strict inequality}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$x<y\hbox{ in }P~\leftrightarrow~x\leq y~\land~x\neq y$$

\item $\leq\,\,:=$ order relation
\item $\leq$ gives rise to $<$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Non-comparability}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$y\not\leq x\not\leq y~\leftrightarrow~x\mid\mid y$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Induced order (inherited)}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$x,y\in Q:x\leq y\hbox{ in }Q~\leftrightarrow~x\leq y\hbox{ in }P$$

\item $P:=$ ordered set;\hspace{0.6cm} $Q\subseteq P$
\item $Q$ has the \texttt{induced order}.
\item $Q$ has the order \texttt{inherited from} $P$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Chain (linearly/totally ordered set), antichain}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(\forall x,y\in P:x\leq y~\veebar~y\leq x)~\equiv~(P:=\texttt{chain})$$

\item \texttt{Def} $$(x\leq y\hbox{ in }P\leftrightarrow x=y)~\equiv~(P:=\texttt{antichain})$$

\item $P:=$ ordered set
\item \texttt{chain} $:=$ \texttt{linearly ordered set} $:=$ \texttt{totally ordered set}
\item Notation: $S:=$ chain; $~\overline S:=$ antichain.

\hrulefill
\item \texttt{Thm} $$\hbox{Any {\bf subset} of a {\bf chain} ({\bf antichain}) is a {\bf chain} ({\bf antichain}).}$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{$n$-element set}

\begin{enumerate}[leftmargin=*,resume]

\item $\textbf{\emph n}=\{0,1,2,...,n-1\}$ is a chain ($\leq$).
\item ${\bf\overline n}:=$ antichain of $\textbf{\emph n}$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Order-isomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def}
\begin{gather*}
\exists\varphi:x\leq y\hbox{ in }P~\leftrightarrow~\varphi(x)\leq\varphi(y)\hbox{ in }Q\\
\equiv~P\cong Q
\end{gather*}

\item $P,Q:=$ ordered sets
\item $(\varphi:P\to Q):=$ map from $P$ onto $Q$
\item $(P\cong Q):=P$ and $Q$ are \texttt{(order-)isomorphic}
\item $\varphi:=$ \texttt{order-isomorphism} (it mirrors the order structure)

\hrulefill
\item \texttt{Thm} $$\varphi\hbox{ is {\bf bijective}.}$$

\item \texttt{Thm} $$(\varphi^{-1}:Q\to P)\equiv\hbox{ order-isomorphism}$$
\item $\varphi^{-1}:=$ \texttt{inverse} of $\varphi$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Number systems}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\mathbb{R},\mathbb{N},\mathbb{N}_0,\mathbb{Q},\mathbb{Z}\equiv \texttt{chains } (\leq)$$

\item $\mathbb N_0:=\mathbb N\cup\{0\}:=\omega$

\hrulefill
\item \texttt{Thm} $$\omega\cong\mathbb N$$
\item $\cong~:=\,\,$ order-isomorphism
\item $(n\mapsto n^+=n+1):=$ successor function from $\mathbb N_0$ to $\mathbb N$ (order-isomorphism)

\hrulefill
\item \texttt{Thm} $$\langle\mathbb N_0,\preceq\rangle\equiv\texttt{order}\hbox{ (not a chain)}$$
\item \texttt{Def}\\ $$m\preceq n~\leftrightarrow~\exists k\in\mathbb N_0:km=n~\hbox{ ($m$ divides $n$)}$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Power set}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\langle{\cal P}(X),\leq\rangle\equiv\texttt{ordered set}$$
\item \texttt{Def} $$A\leq B~\leftrightarrow~A\subseteq B$$

\item $X:=$ set
\item ${\cal P}(X):=$ power set of $X$

\hrulefill
\item \texttt{Thm} $$\forall Y\subseteq{\cal P}(X):Y\hbox{ is ordered }(\subseteq)$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Sub}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm:} {\it The set of all (subgroups, normal subgroups, subspaces of a vector space, subrings, ideals of a ring) are ordered under set inclusion.}

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Topological space}

\begin{enumerate}[leftmargin=*,resume]

\item $(X,{\cal T}):=$ topological space
\item Families of open, closed, and clopen subsets of $X$ can be ordered under set inclusion.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Predicate, power set, order-isomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\langle{\cal P}(X),\subseteq\rangle,\,\,\langle\mathbb P,\Rrightarrow\rangle\equiv\texttt{ordered sets}$$

\item \texttt{Def} $$p,q\in\mathbb P(X):p\Rrightarrow q~\hbox{ iff }~\{x\in X\mid p(x)=T\}\subseteq\{x\in X\mid q(x)=T\}$$

\item $(p:X\to\{T,F\}):=$ predicate (statement true or false)
\item $T:=$ true, $~F:=$ false
\item $\mathbb P(X):=$ set of predicates on X
\item ${\cal P}(X):=$ power set of X

\item $\Rrightarrow~:=~$ order relation (implication)

\hrulefill
\item \texttt{Thm} $$(\varphi:\mathbb P(X)\to{\cal P}(X))~\equiv~\texttt{order-isomorphism}$$
\item $\varphi(p):=\{x\in X\mid p(x)=T\}$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Covering}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def}
\begin{gather*}
x<y~\land~x\leq z<y~\to~z=x\\
\equiv~x\hbox{ is }\texttt{covered by }y\hbox{ (or }y\texttt{ covers }x\hbox{)}\\
\equiv~x\covering y~\equiv~y\covers x
\end{gather*}

\item $P:=$ ordered set;\hspace{0.6cm} $x,y\in P$
\item $\not\exists z\in P:x<z<y$ when $x\covering y$.
\item $\leq,\,<\,\,:=$ order relations
\item $\covering,\,\covers\,\,:=$ \texttt{covering relations}

\hrulefill
\item \texttt{Thm} $$(P\hbox{ is finite})~\leftrightarrow~(x<y~\leftrightarrow~\exists s:s=(x=x_0\covering x_1\covering\hdots\covering x_n=y))$$

\item For finite $P$, $<$ determines (and is determined by) $\covering$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Diagrams}

\begin{enumerate}[leftmargin=*,resume]

\item Check pp. 11-13 of \cite{priestley}.
\item $x<y$ iff there are connected line segments moving upwards from $x$ to $y$
\item diagram of ${\cal P}(\{1,2,3\}):=$ the cube

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Order-isomorphism, covering}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\varphi\hbox{ is an }\texttt{order-isomorphism}$$ $$\equiv$$ $$x\covering y\hbox{ in }P~\hbox{ iff }~\varphi(x)\covering\varphi(y)\hbox{ in }Q$$

\item $P,Q:=$ {\it finite} ordered sets
\item $(\varphi:P\to Q):=$ bijective map

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Order-isomorphism, identical diagrams}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$P\cong Q~\hbox{ iff }~P,Q\hbox{ have identical diagrams}$$

\item $P,Q:=$ {\it finite} ordered sets
\item $\cong~:=\,\,$ order-isomorphism

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Dual}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$x\leq y\hbox{ in }P^\partial~\leftrightarrow~y\leq x\hbox{ in }P$$

\item $P:=$ ordered set
\item $P^\partial:=$ dual of $P$
\item diagram for $P^\partial=$ diagram for $P$ (upside down)

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Duality Principle}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\forall P:\Phi~\leftrightarrow~\forall P:\Phi^\partial$$

\item $P:=$ ordered set
\item $\Phi:=$ true statement in $P$
\item $\Phi^\partial:=$ true dual statement in $P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Bottom, Top}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$(\exists\bot\in P:\forall x\in P,~\bot\leq x)~\equiv~P\hbox{ has a }\texttt{bottom}\textit{ element}$$

\item\texttt{Def} $$(\exists\top\in P:\forall x\in P,~x\leq\top)~\equiv~P\hbox{ has a }\texttt{top}\textit{ element}$$

\item $P:=$ ordered set
\item $\bot:=$ \texttt{bottom}
\item $\top:=$ \texttt{top}

\hrulefill
\item \texttt{Thm}\label{e1} $$(\exists!\bot\in P)\hbox{ is dual to ($\exists!\top\in P$).}$$
\item (\ref{e1}) is {\it an instance of the duality theorem}.

\hrulefill
\item An infinite chain may not have bottom or top elements.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lifting}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{lifting} $:=$ adding a bottom element to ordered sets

\item \texttt{Def} $$x\leq y~\hbox{ iff }~x={\bf0}\hbox{ or }x\leq y\hbox{ in }P$$
\item ${\bf0}\not\in P$, \hspace{0.3cm} $P_\bot:=P\cup\{{\bf0}\}$

\hrulefill

\item \texttt{Thm} $$\forall S:\overline S_\bot\hbox{ is an }\texttt{ordered set}$$

\item $S:=$ set
\item $\overline S_\bot:=$ \texttt{flat} ordered set (lifted antichain)

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Maximal, minimal, greatest (maximum),\\ least (minimum)}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} $$(a\leq x~\land~x\in Q)\to a=x$$$$\equiv~a\in Q\hbox{ is a }\texttt{maximal element of }Q$$

\item $P:=$ ordered set;\hspace{0.6cm} $Q\subseteq P$;\hspace{0.6cm} $a,x\in Q$

\item A maximal element is greater than all comparable elements in $P$, different from itself.

\hrulefill
\item \texttt{Thm} $$(Q\hbox{ has the order inherited from }P,~\exists\top_Q)~\to~\hbox{Max}\,Q=\{\top_Q\}$$

\item $\top_Q:=$ top element of $Q:=$ max$\,Q$\\ $:=$ \texttt{greatest} (or \texttt{maximum}) element of $Q$
\item Max$\,Q:=$ set of \texttt{maximal} elements of $Q$

\hrulefill
\item \texttt{Thm} $$(Q\hbox{ has the order inherited from }P,~\exists\bot_Q)~\to~\hbox{Min}\,Q=\{\bot_Q\}$$

\item $\bot_Q:=$ bottom element of $Q:=$ min$\,Q$\\ $:=$ \texttt{least} (or \texttt{minimum}) element of $Q$
\item Min$\,Q:=$ set of \texttt{minimal} elements of $Q$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Disjoint union, linear sum, $M_n$}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$P~\dot\cup~Q\hbox{ is an }\texttt{ordered set}:$$ $$x\leq y\hbox{ in }P~\dot\cup~Q~\hbox{ iff }~(x,y\in P,~x\leq y\hbox{ in }P)~\veebar~(x,y\in Q,~x\leq y\hbox{ in }Q)$$

\item $P\cap Q=\emptyset$
\item $P~\dot\cup~Q:=$ \texttt{disjoint union} of $P$ and $Q$

\hrulefill
\item \texttt{Def}
\begin{gather*}
P\oplus Q:=\texttt{linear sum}\\
(\hbox{defined by the following order relation on }P\cup Q)\\
x\leq y\hbox{ iff }(x,y\in P,~x\leq y\hbox{ in }P)\lor(x,y\in Q,~x\leq y\hbox{ in }Q)\lor(x\in P,~y\in Q)\\
\end{gather*}

\item $P,Q:=$ disjoint ordered sets

\hrulefill
\item $P_\bot:={\bf0}\oplus P~$ (lifting)
\item $P\oplus{\bf1}~$ ($P$ with a new top element)

\hrulefill
\item \texttt{Thm} (associativity) $$P~\dot\cup~(Q~\dot\cup~R)=(P~\dot\cup~Q)~\dot\cup~R$$ $$P\oplus(Q\oplus R)=(P\oplus Q)\oplus R$$

\hrulefill
\item \texttt{Def} $${\bf M_n}={\bf1}\oplus{\bf \overline n}\oplus{\bf1}$$

\item $\textbf{\emph n}=\{0,1,2,...,n-1\}$ is a chain ($\leq$).
\item ${\bf\overline n}:=$ antichain of $\textbf{\emph n}$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Products}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def:} $P_1\times\hdots\times P_n$ is an \texttt{ordered set} if  the \texttt{coordinatewise order} holds, $$(x_1,...,x_n)\leq(y_1,...,y_n)~\leftrightarrow~\forall i:x_i\leq y_i\hbox{ in }P_i.$$

\item $P_1,...,P_n:=$ ordered sets

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lexicographic order}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(x_1,x_2)\leq(y_1,y_2)~\hbox{ if }~x_1<y_1~\lor~(x_1=y_1,\,\,x_2\leq y_2)$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Down-set, up-set}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def}
\begin{gather*}
x\in Q,~y\in P,~y\leq x~\to~y\in Q\\
(Q:=\texttt{down-set}:=\texttt{decreasing set}:=\texttt{order ideal})
\end{gather*}

\item \texttt{Def}
\begin{gather*}
x\in Q,~y\in P,~y\geq x~\to~y\in Q\\
(Q:=\texttt{up-set}:=\texttt{increasing set}:=\texttt{order filter})
\end{gather*}

\item $P:=$ ordered set,\hspace{0.3cm} $Q\subseteq P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Principal}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$\downarrow Q:=\{y\in P\mid\exists x\in Q:y\leq x\}$$
\item \texttt{Def} $$\uparrow Q:=\{y\in P\mid\exists x\in Q:y\geq x\}$$

\item \texttt{Def} $$\downarrow x:=\{y\in P\mid y\leq x\}$$
\item \texttt{Def} $$\uparrow x:=\{y\in P\mid y\geq x\}$$

\item $P:=$ ordered set,\hspace{0.5cm} $Q\subseteq P$,\hspace{0.5cm} $x\in P$
\item $\downarrow Q,\,\,\downarrow x:=$ down-sets
\item $\uparrow Q,\,\,\uparrow x:=$ up-sets

\item $\downarrow Q$ reads {\it down Q};\hspace{0.5cm} $\uparrow Q$ reads {\it up Q}.

\hrulefill
\item $\downarrow Q$ is the {\bf smallest down-set} containing $Q$.

\hrulefill
\item \texttt{Thm} $$Q\equiv\texttt{down-set}~\leftrightarrow~Q=\,\,\downarrow Q$$
\item \texttt{Thm} $$Q\equiv\texttt{up-set}~\leftrightarrow~Q=\,\,\uparrow Q$$

\item \texttt{Thm} $$\downarrow\{x\}=\,\,\downarrow x;\hspace{0.77cm}\uparrow\{x\}=\,\,\uparrow x$$

\item $\downarrow x,\,\uparrow x:=$ \texttt{principals}

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Family of down-sets, order-isomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\langle{\cal O},\subseteq\rangle\hbox{ is an }\texttt{ordered set}.$$

\item $P:=$ ordered set
\item ${\cal O}(P):=$ set of all \texttt{down-sets} of $P$

\hrulefill
\item \texttt{Thm} $$P\hbox{ is an }\texttt{antichain}~\to~{\cal O}(P)={\cal P}(P)$$
\item ${\cal P}(X):=$ power set of $X$

\hrulefill
\item \texttt{Thm} $$Q\hbox{ is a }\texttt{down-set}~\leftrightarrow~P\backslash Q\hbox{ is an }\texttt{up-set}$$

\item $\backslash~:=\,\,$ set complementation (``subtraction'' of sets)

\hrulefill
\item \texttt{Thm} $${\cal O}(P)^\partial~\cong~{\cal O}(P^\partial)$$ $$\hbox{(complementation map)}$$

\item $\partial~:=\,\,$ dual
\item $\cong~:=\,\,$ order-isomorphism

\hrulefill
\item \texttt{Thm} $${\cal O}(P\oplus{\bf1})\cong{\cal O}(P)\oplus{\bf1}$$ $${\cal O}({\bf1}\oplus P)\cong{\bf1}\oplus{\cal O}(P)$$ $${\cal O}(P_1\dot\cup P_2)\cong{\cal }{\cal O}(P_1)\times{\cal }{\cal O}(P_2)$$

\hrulefill
\item \texttt{Thm} $${\cal O}(P_1\dot\cup P_2)~\cong_U~{\cal O}(P_1)\times{\cal O}(P_2)$$ $$U\mapsto(U\cap P_1,U\cap P_2)$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Ordered relation, down-sets}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$x\leq y~\equiv~\downarrow x\subseteq\,\downarrow y~\equiv~\forall Q\in{\cal O}(P):y\in Q\to x\in Q$$

\item $P:=$ ordered set, \hspace{0.3cm} $x,y\in P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Order-preserving/embedding}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(x\leq_P y\to\varphi(x)\leq_Q\varphi(y))~\to~\varphi:=\texttt{order-preserving (monotone)}$$ $$(x\leq_P y\leftrightarrow\varphi(x)\leq_Q\varphi(y))~\to~(\varphi:P\hookrightarrow Q):=\texttt{order-embedding}$$

\item $P,Q:=$ ordered-sets,\hspace{0.5cm} $\varphi:P\to Q$
\item order embedding (one-to-one)

\hrulefill
\item \texttt{Thm} $$\varphi(P)\cong P$$

\item $(\varphi:P\hookrightarrow Q):=$ \texttt{order-embedding} map
\item $\varphi(P)=\{\varphi(x)\mid x\in P\}$
\item $\cong~:=\,\,$ order-isomorphism
\item $\varphi(P)$ is {\it embedded} in $P$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Composite, order-preserving maps}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm:} {\it The composite of a finite number of order-preserving maps is order-preserving.}

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Pointwise order}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def:} \texttt{pointwise order} $$f\leq g~\equiv~\forall x\in\mathbb R:f(x)\leq g(x)$$

\item $f,g:\mathbb R\to\mathbb R$
\item $\leq~:=\,\,$ order relation of maps (functions)

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Upper bound, lower bound}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(\forall s\in S:s\leq x)~\to~(x\in P):=\texttt{upper bound}\hbox{ of }S$$

\item \texttt{Def} $$(\forall s\in S:s\geq x)~\to~(x\in P):=\texttt{lower bound}\hbox{ of }S$$

\item \texttt{Def} $$S^u:=\{x\in P\mid\forall s\in S:s\leq x\}$$
\item \texttt{Def} $$S^\ell:=\{x\in P\mid\forall s\in S:s\geq x\}$$

\item $P:=$ ordered set;\hspace{0.5cm} $S\subseteq P$
\item $S^u:=$ set of all upper bounds of $S$
\item $S^\ell:=$ set of all lower bounds of $S$
\item $S^u$ reads {\it S upper} and $S^\ell$ reads {\it S lower}.

\hrulefill
\item \texttt{Thm} $$S^u\hbox{ is an up-set}.$$
\item \texttt{Thm} $$S^\ell\hbox{ is a down-set}.$$

\hrulefill
\item \texttt{Thm} $$\exists_{\leq1}\,a,b:a=\ell ub,\,b=glb$$

\item $\ell ub:=$ \texttt{least upper bound (supremum)} $:=$ sup$\,S$
\item $glb:=$ \texttt{greatest lower bound (infimum)} $:=$ inf$\,S$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Join, meet}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$(x\lor y):=\hbox{sup}\{x,y\}$$
\item \texttt{Def} $$(x\land y):=\hbox{inf}\{x,y\}$$

\item \texttt{Def} $$\bigvee S:=\hbox{sup}\,S$$
\item \texttt{Def} $$\bigwedge S:=\hbox{inf}\,S$$

\item $(x\lor y):=x$ \texttt{join} $y:=x$ \texttt{supremum} $y$
\item $(x\land y):=x$ \texttt{meet} $y:=x$ \texttt{infimum} $y$
\item $\bigvee S:=$ \texttt{join of} $S:=$ \texttt{supremum}\hbox{ of} $S$
\item $\bigwedge S:=$ \texttt{meet of} $S:=$ \texttt{infimum}\hbox{ of} $S$
\item $P:=$ ordered set
\item $\bigvee_P S:=$ \texttt{join} of $S$ in $P:=$ \texttt{supremum}\hbox{ of} $S$ in $P$
\item $\bigwedge_P S:=$ \texttt{meet} of $S$ in $P:=$ \texttt{infimum}\hbox{ of} $S$ in $P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lattice, complete lattice}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$\forall x,y\in P~\exists j,m\in P:j=x\lor y,\,m=x\land y$$ $$(P:=\texttt{lattice})$$

\item \texttt{Def} $$\forall S\subseteq P~\exists j,m:j=\bigvee S,\,m=\bigwedge S$$ $$(P:=\texttt{complete lattice})$$

\item $\emptyset\neq P:=$ ordered set
\item $(x\lor y):=x$ \texttt{join} $y$
\item $(x\land y):=x$ \texttt{meet} $y$
\item $\bigvee S:=$ \texttt{join of} $S$
\item $\bigwedge S:=$ \texttt{meet of} $S$

\hrulefill
\item\texttt{Thm}
\begin{gather*}
\forall a,b,c,d\in P:\\
a\leq b~\to~a\lor c\leq b\lor c,~a\land c\leq b\land c,\\
a\leq b,~c\leq d~\to~a\lor c\leq b\lor d,~a\land c\leq b\land d
\end{gather*}

\item $P:=$ lattice

\hrulefill
\item \texttt{Thm} $$x\leq y~\to~x\lor y=y,~x\land y=x$$

\item $\emptyset\neq P:=$ ordered-set

\hrulefill
\item \texttt{Thm}
\begin{gather*}
\hbox{To prove that }P\hbox{ is a lattice is equivalent to prove}\\
\forall x,y\in P\hbox{ with }x\mid\mid y:\exists j,m:j=x\lor y,\,m=x\land y.
\end{gather*}

\item $\emptyset\neq P:=$ ordered-set

\hrulefill
\item\texttt{Thm} $$\{x,y\}^u=\uparrow y,\hspace{0.6cm}\{x,y\}^\ell=\downarrow y$$
\item $x,y\in (P:=$ ordered set$)$,\hspace{0.5cm} $x\leq y$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Power set, complete lattice}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\langle{\cal P}(X),\subseteq\rangle\hbox{ is a }\texttt{complete lattice}$$

\item \texttt{Def}
\begin{gather*}
\bigvee A_i=\bigcup A_i\\
\bigwedge A_i=\bigcap A_i
\end{gather*}

\item ${\cal P}(X):=$ power set of $X$
\item $\bigvee A_i:=\bigvee\{A_i\mid i\in I\}$
\item $\bigcup A_i:=\bigcup\{A_i\mid i\in I\}$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lattice of sets}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$\mathfrak L\hbox{ is {\it closed} under }\texttt{finite unions}\hbox{ and }\texttt{intersections}~\to$$ $$\to~\mathfrak L\hbox{ is a }\texttt{lattice of sets}$$

\item \texttt{Def} $$\mathfrak L\hbox{ is {\it closed} under }\texttt{arbitrary unions}\hbox{ and }\texttt{intersections}~\to$$ $$\to~\mathfrak L\hbox{ is a }\texttt{complete lattice of sets}$$

\item $\emptyset\neq\mathfrak L\subseteq{\cal P}(X)$
\item ${\cal P}(X):=$ power set of $X$

\hrulefill
\item \texttt{Def}
\begin{gather*}
\mathfrak L\hbox{ is a lattice of sets}~\to~\langle\mathfrak L,\subseteq\rangle\hbox{ is a lattice},\\
A\lor B=A\cup B,~A\land B=A\cap B
\end{gather*}

\hrulefill
\item\texttt{Thm}
\begin{gather*}
{\cal O}(P)\hbox{ is a }\texttt{complete lattice of sets}\\
(\textbf{down-set lattice}\hbox{ of }P).
\end{gather*}

\item $P:=$ ordered set
\item ${\cal O}(P):=$ ordered set of all down-sets of $P$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{$M_n$, lattice}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $${\bf M_n}\hbox{ is a }\texttt{lattice}.$$

\item ${\bf M_n}={\bf1}\oplus{\bf \overline n}\oplus{\bf1}$ (ordered set for $n\geq1$)

\item \texttt{Def}
\begin{gather*}
P\oplus Q:=\texttt{linear sum}\\
x\leq y\hbox{ iff }(x,y\in P,~x\leq y\hbox{ in }P)\lor(x,y\in Q,~x\leq y\hbox{ in }Q)\lor(x\in P,~y\in Q)
\end{gather*}

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Ordered by division}

\begin{enumerate}[leftmargin=*,resume]
\item \texttt{Thm}
\begin{gather*}
\langle\mathbb N_0,\preceq\rangle\hbox{ is a }\texttt{lattice},\\
m\lor n=\hbox{lcm}\{m,n\},~m\land n=\hbox{gcd}\{m,n\}
\end{gather*}

\item \texttt{Def}\\ $$m\preceq n~\leftrightarrow~\exists k\in\mathbb N_0:km=n~\hbox{ ($m$ divides $n$)}$$

\item lcm $:=$ least common multiple
\item gcd $:=$ greatest common divisor

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lattice, algebraic structure}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def} $$a\lor b:=\hbox{sup}\{a,b\},~~a\land b:=\hbox{inf}\{a,b\}$$

\item $a,b\in (L:=$ lattice$)$
\item $\lor,\land:L^2\to L$
\item $\langle L,\lor,\land\rangle:=$ algebraic structure

\item \texttt{Thm} $$\lor,\land\hbox{ are }\texttt{order preserving}.$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Connecting Lemma}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$a\leq b~\equiv~a\lor b=b~\equiv~a\land b=a$$

\item $a,b\in (L:=$ lattice$)$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Associativity, commutativity, idempotency,\\ absorption}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm:} (Associativity) $\forall a,b,c\in L$
\begin{align*}
&(a\lor b)\lor c=a\lor(b\lor c)\\
&(a\land b)\land c=a\land(b\land c)
\end{align*}

\item \texttt{Thm:} (Commutativity) $\forall a,b,c\in L$
\begin{align*}
&a\lor b=b\lor a\\
&a\land b=b\land a
\end{align*}

\item \texttt{Thm:} (Idempotency) $\forall a,b,c\in L$
\begin{align*}
&a\lor a=a\lor a\\
&a\land a=a\land a
\end{align*}

\item \texttt{Thm:} (Absorption) $\forall a,b,c\in L$
\begin{align*}
&a\lor(a\land b)=a\\
&a\land(a\lor b)=a
\end{align*}

\item $L:=$ lattice

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Join, meet, order relation, algebraic structure}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$\forall a,b\in L:a\lor b=b\leftrightarrow a\land b=a$$

\item \texttt{Thm} $$(a\lor b=b\to a\leq b)~\to~(\leq\hbox{ is an }\texttt{order relation})$$

\item \texttt{Thm}
\begin{gather*}
(a\lor b=b\to a\leq b)~\to~\langle L,\leq\rangle\hbox{ is a }\texttt{lattice},\\
\forall a,b\in L:a\lor b=\hbox{sup}\{a,b\},~a\land b=\hbox{inf}\{a,b\}
\end{gather*}

\item $\langle L,\lor,\land\rangle:=$ algebraic structure
\item $\emptyset\neq L:=$ lattice
\item $\lor,\land:=$ binary operations

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Sublattice}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Def}
\begin{gather*}
\hbox{if }a,b\in M\to (a\lor b\in M,a\land b\in M)\\
\hbox{then }M\hbox{ is a }\texttt{sublattice}\hbox{ of }L
\end{gather*}

\item $L:=$ lattice
\item $\emptyset\neq M\subseteq L$
\item Sub$\,L:=$ collection of all sublattices of $L$
\item Sub$_0\,L=$ Sub$\,L\cup\{\emptyset\}$
\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Product of lattices}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} $$L\times K\hbox{ is a }\texttt{lattice}.$$

\item \texttt{Def}
\begin{gather*}
(\ell_1,k_1)\lor(\ell_2,k_2)=(\ell_1\lor\ell_2,k_1\lor k_2),\\
(\ell_1,k_1)\land(\ell_2,k_2)=(\ell_1\land\ell_2,k_1\land k_2)
\end{gather*}

\item $L,K:=$ lattices

\item \texttt{Thm} $$(\ell_1,k_1)\lor(\ell_2,k_2)=(\ell_2,k_2)~\leftrightarrow~(\ell_1,k_1)\leq(\ell_2,k_2)\\
$$

\item $(\ell_1,k_1)\leq(\ell_2,k_2)$ means $\ell_1\leq\ell_2$ and $k_1\leq k_2$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lattice Homomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm}
\begin{gather*}
f\hbox{ is a }\texttt{lattice homomorphism}\hbox{ if }\forall a,b\in L:\\
f(a\lor b)=f(a)\lor f(b),\hspace{0.3cm}\textbf{join-preserving},\\
f(a\land b)=f(a)\land f(b),\hspace{0.3cm}\textbf{meet-preserving}
\end{gather*}

\item \texttt{Def} $$\texttt{lattice isomorphism}:=\hbox{bijective homomorphism}$$

\item \texttt{Def}
\begin{gather*}
f\hbox{ is a }\texttt{one-to-one homomorphism}~\to\\
\to~(\texttt{sublattice }f(L)\hbox{ of }K)~\cong~L\\
(f\hbox{ is an }\textbf{embedding of $L$ into $K$})
\end{gather*}

\item $L,K:=$ lattices
\item $f:L\to K$
\item $\cong~:=\,\,$ isomorphism

\hrulefill
\item \texttt{Thm} $$(M\rightarrowtail L)~\to~(M\hookrightarrow L)$$
\item $(M\rightarrowtail L):=$ sublattice of $L$ $\cong M$
\item $(M\hookrightarrow L):=$ order-embedding

\hrulefill
\item \texttt{Def} $$f:=\textbf{ lattice }\{0,1\}\hbox{-}\textbf{homomorphism}$$
\item $L,K:=$ bounded lattices
\item $f:L\to K$
\item $f(0)=0,~~f(1)=1$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Map, order-preserving, homomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item \texttt{Thm} 
\begin{gather*}
f\hbox{ is }\texttt{order-preserving}~\equiv\\
\equiv~\forall a,b\in L:f(a\lor b)\geq f(a)\lor f(b)\\
\equiv~\forall a,b\in L:f(a\land b)\leq f(a)\land f(b)
\end{gather*}

\item \texttt{Thm} $$f\hbox{ is a }\texttt{homomorphism}~\to~f\hbox{ is }\texttt{order-preserving}$$

\item $L,K:=$ lattices
\item $f:L\to K$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Lattice/order-isomorphism}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm} $$A\cong_{\ell att}B~~\hbox{ iff }~~A\cong_\leq B$$

\item $A,B:=$ lattices
\item $\cong_{\ell att}~:=\,\,$ lattice isomorphism
\item $\cong_\leq~:=\,\,$ order-isomorphism

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Ideals, filters}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm}
\begin{gather*}
J\hbox{ is an }\texttt{ideal}\hbox{ if}\\
a,b\in J~\to~a\lor b\in J\\
a\in L,\,b\in J,\,a\leq b~\to~a\in J
\end{gather*}

\item\texttt{Thm}
\begin{gather*}
G\hbox{ is a }\texttt{filter}\hbox{ if}\\
a,b\in G~\to~a\land b\in G\\
a\in L,\,b\in G,\,a\geq b~\to~a\in G
\end{gather*}

\item $L:=$ lattice
\item $\emptyset\neq J\subseteq L$,\hspace{0.5cm} $\emptyset\neq G\subseteq L$
\item\texttt{ideal} $:=$ non-empty down-set closed under join
\item\texttt{filter} $:=$ order dual of lattice ideal
\item ${\cal I}(L),~{\cal F}(L):=$ set of all ideals and filters of $L$, respectively ($\subseteq$)

\hrulefill
\item\texttt{Thm:} $$\forall J:J\hbox{ is a }\texttt{sublattice}\hbox{ of }L,\hbox{ i.e., }\forall a,b\in L:a\land b\leq a$$
\item $J:=$ ideal of $L$

\hrulefill
\item\texttt{Thm} $$J\hbox{ is }\texttt{proper}~\hbox{ iff }~1\not\in J$$
\item\texttt{Thm} $$G\hbox{ is }\texttt{proper}~\hbox{ iff }~0\not\in G$$

\item $J:=$ \texttt{ideal} of $L$ with 1 (top)
\item $G:=$ \texttt{filter} of $L$ with 0 (bottom)

\hrulefill
\item\texttt{Thm}
\begin{gather*}
\forall a\in L:\,\,\,\downarrow a\hbox{ is an }\texttt{ideal}\\
\hbox{(\texttt{principal ideal} generated by }a\hbox{)}
\end{gather*}

\item\texttt{Thm}
\begin{gather*}
\forall a\in L:\,\,\,\uparrow a\hbox{ is a }\texttt{filter}\\
\hbox{(\texttt{principal filter} generated by }a\hbox{)}
\end{gather*}

\hrulefill
\item\texttt{Thm} $$\forall J,G:J,G\hbox{ are }\texttt{principal}$$ $$J=\,\,\downarrow\bigvee J$$ $$G=\,\,\uparrow\bigwedge G$$

\item $J:=$ ideal of the {\it finite} lattice $L$
\item $G:=$ filter of the {\it finite} lattice $L$

\hrulefill
\item\texttt{Thm} $$f^{-1}(0)\hbox{ is an }\texttt{ideal}$$ $$f^{-1}(1)\hbox{ is a }\texttt{filter}$$

\item $L,K:=$ bounded lattices
\item $(f:L\to K):=\{0,1\}$-homomorphism

\hrulefill
\item\texttt{Thm} $$G\hbox{ is a }\texttt{filter}\hbox{ in }{\cal P}(X).$$

\item $(X,{\cal T}):=$ topological space
\item $x\in X$
\item $G=\{V\subseteq X\mid \exists U\in{\cal T}:x\in U\subseteq V\}$
\item ${\cal P}(X):=$ power set of $X$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Join, meet, two subsets, ordered sets}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm}
\begin{align*}
(i)&~~\forall s\in S:s\leq\bigvee S,~s\geq\bigwedge S\\
(ii)&~~x\leq\bigwedge S~\leftrightarrow~\forall s\in S:x\leq S\\
(iii)&~~x\geq\bigvee S~\leftrightarrow~\forall s\in S:x\geq S\\
(iv)&~~\bigvee S\leq\bigwedge T~\leftrightarrow~\forall s\in S~\forall t\in T:s\leq t\\
(v)&~~S\subseteq T~\to~\bigvee S\leq\bigvee T,~\bigwedge S\geq\bigwedge T
\end{align*}

\item $P:=$ ordered set
\item $S,T\subseteq P$
\item $\bigvee S:=$ join of $S$,\hspace{0.6cm} $\bigwedge T:=$ meet of $T$
\item $\bigvee S,\bigvee T,\bigwedge S,\bigwedge T$ exist.
\item $x\in P$

\hrulefill
\item\texttt{Thm} $$\bigvee(S\cup T)=(\bigvee S)\lor(\bigvee T)$$ $$\bigwedge(S\cup T)=(\bigwedge S)\land(\bigwedge T)$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Existence, join, meet, finite subset, lattice}

\begin{enumerate}[leftmargin=*,resume]

\item $$\forall F\subseteq P~\exists s,t:s=\bigvee F,~t=\bigwedge F$$

\item $P:=$ lattice
\item $F:=$ finite, non-empty subset of $P$
\item $\bigvee F:=$ join of $F$,\hspace{0.6cm} $\bigwedge F:=$ meet of $F$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Finite lattice complete}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm:} {\it Every finite lattice is complete.}

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Preservation, existing joins and meets}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def:} $\varphi$ \texttt{preserves existing joins} if
\begin{align*}
\exists j\in P:j=\bigvee S~\to~\exists j'\in Q:j'=\bigvee\varphi(S),\\
\varphi(\bigvee S)=\bigvee\varphi(S)
\end{align*}

\item\texttt{Def:} $\varphi$ \texttt{preserves existing meets} if
\begin{align*}
\exists m\in P:m=\bigwedge S~\to~\exists m'\in Q:m'=\bigwedge\varphi(S),\\
\varphi(\bigwedge S)=\bigwedge\varphi(S)
\end{align*}

\item $P,Q:=$ ordered sets
\item $\varphi:P\to Q$

\hrulefill
\item\texttt{Thm} $$\exists j\in P:j=\bigvee S,~\exists j'\in Q:j'=\bigvee\varphi(S)~\to~\varphi(\bigvee S)\geq\bigvee\varphi(S)$$ $$\exists m\in P:m=\bigwedge S,~\exists m'\in Q:m'=\bigwedge\varphi(S)~\to~\varphi(\bigwedge S)\geq\bigwedge\varphi(S)$$

\item $(\varphi:P\to Q):=$ order-preserving map
\item $S\subseteq P$

\hrulefill
\item\texttt{Thm} $$\varphi\textit{ preserves all}\texttt{ joins }\hbox{and}\texttt{ meets}.$$

\item $(\varphi:P\to Q):=$ order-isomorphism

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Subset induced order} % 2.28

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm} $$\exists j\in Q:j=\bigvee_PS~\to~\bigvee_QS=\bigvee_PS$$ $$\exists m\in Q:m=\bigwedge_PS~\to~\bigwedge_QS=\bigwedge_PS$$

\item $P:=$ ordered set
\item $S\subseteq Q\subseteq P$
\item $Q$ has the order inherited from $P$.

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Complete lattice, equivalence}

\begin{enumerate}[leftmargin=*,resume]

\item
\begin{align*}
P\hbox{ is a }\texttt{complete lattice}\\
\equiv~\forall S\subseteq P~\exists m\in P:m=\bigwedge S\\
\equiv~\top\in P,~\forall S_{\neq\emptyset}\subseteq P~\exists m\in P:m=\bigwedge S
\end{align*}

\item $\emptyset\neq P:=$ ordered set
\item $\top:=$ top element

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Subset power set, complete lattice inclusion}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm} 
\begin{align*}
\left(\forall\{A_i\}\seq\cll:\bigcap_iA_i\in\cll\right)~\land~(X\in\cll)~\to\\
\to~\cll\hbox{ is a }\texttt{complete lattice:}\\
\bigwedge_iA_i&=\bigcap_iA_i,\\
\bigvee_iA_i&=\bigcap\{B\in\cll\mid\bigcup_iA_i\seq B\}
\end{align*}

\item $X:=$ set,\hspace{0.5cm} $i\in I$,\hspace{0.5cm} $\{A_i\}\neq\emptyset$
\item $\cll:=$ family of subsets of $X$
\item $(\cll,\seq):=$ ordered set

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{(Topped) Intersection structure}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def}
\begin{align*}
\forall\{A_i\}\seq\cll:\bigcap_iA_i\in\cll~\to\\
\to~\cll\hbox{ is an }\texttt{intersection structure}\hbox{ }(\bigcap\texttt{-structure})\hbox{ on }X
\end{align*}

\item\texttt{Def}
\begin{align*}
\left(\forall\{A_i\}\seq\cll:\bigcap_iA_i\in\cll\right)~\land~(X\in\cll)~\to\\
\to~\cll\hbox{ is a }\texttt{topped intersection structure}\hbox{ on }X\\
\texttt{(closure system)}
\end{align*}

\item $\emptyset\neq\cll:=$ family of subsets of $X$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Join/meet-irreducibility}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Def} 
\begin{gather*}
x\in L\hbox{ is }\texttt{join-irreducible}\hbox{ if }\\
(x\neq0)\hbox{ and }(\forall a,b\in L:~x=a\lor b~\to~x=a\hbox{ or }x=b)
\end{gather*}

\item\texttt{Def} 
\begin{gather*}
x\in L\hbox{ is }\texttt{meet-irreducible}\hbox{ if }\\
(x\neq1)\hbox{ and }(\forall a,b\in L:~x=a\land b~\to~x=a\hbox{ or }x=b)
\end{gather*}

\item $L:=$ lattice

\item \texttt{join-irreducible element} $:=$ {\it cannot} be expressed as the \texttt{join} of any other elements in the lattice

\item \texttt{meet-irreducible element} $:=$ {\it cannot} be expressed as the \texttt{meet} of any other elements in the lattice

\hrulefill
\item ${\cal J}(L):=$ set of join-irreducible elements of $L$
\item ${\cal M}(L):=$ set of meet-irreducible elements of $L$
\item ${\cal J}(L)$ and ${\cal M}(L)$ inherit $L$'s order relation.

\hrulefill
\item\texttt{Def}
\begin{gather*}
\forall a\in P\,\,\exists A\seq Q:a=\bigvee_PA~\to~Q\hbox{ is }\texttt{join-dense}\hbox{ in }P
\end{gather*}

\item\texttt{Def}
\begin{gather*}
\forall a\in P\,\,\exists A\seq Q:a=\bigwedge_PA~\to~Q\hbox{ is }\texttt{meet-dense}\hbox{ in }P
\end{gather*}

\item $P:=$ ordered set
\item $Q\seq P$

\hrulefill
\item\texttt{Thm} $$\textit{In a finite lattice every element is a join of join-irreducible elements.}$$

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Algebraic theory of lattices}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm} $$(L,\leq)\dashv\vdash(L,\lor,\land)$$

\item $(L,\leq):=$ lattice (order relation)
\item $(L,\lor,\land):=$ lattice (algebraic structure)

\end{enumerate}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Equivalence join meet}

\begin{enumerate}[leftmargin=*,resume]

\item\texttt{Thm}
\begin{eqnarray*}
(x\land y)\lor(x\land z)&\leq &x\land(y\lor z)\\
x\lor(y\land z)&\leq &(x\lor y)\land(x\lor z)\\
(x\land y)\lor(y\land z)\lor(z\land x)&\leq &(x\lor y)\land(y\lor z)\land(z\lor x)\\
(x\land y)\lor(x\land z)&\leq &x\land(y\lor(x\land z))
\end{eqnarray*}

\item $x,y,z\in(L:=$ lattice$)$

\end{enumerate}



\newpage
%**
%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Open Invitation}

\noindent {\it Review}, {\it add} content, and \texttt{co-author} this white paper \cite{microarticles,ojmp-guidelines}.\\ {\it Join} the {\bf Open Mathematics Collaboration}.\\ Send your contribution to \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}}.

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Supplementary files}

The {\bf latex file} for this {\it white paper} together with other {\it supplementary files} are available in \cite{ojmp-project,zen}.

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{How to cite this paper?}

\noindent\url{https://doi.org/10.31219/osf.io/fzpc4}\\
\noindent\url{https://zenodo.org/record/7267788}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Acknowledgements}

\begin{itemize}[leftmargin=*]

\item[$+$] {\bf Center for Open Science}\\\url{https://cos.io}
\item[$+$] {\bf Open Science Framework}\\\url{https://osf.io}
\item[$+$] {\bf Zenodo}\\\url{https://zenodo.org}

\end{itemize}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{Agreement}

\noindent The author agrees with \cite{ojmp-guidelines}.
%\noindent All authors agree with \cite{ojmp-guidelines}.

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\section*{License}

\noindent {\it CC-By Attribution 4.0 International} \cite{cc4}

%––––––––––––––––––––––––––––––––––––––––––––––––––––––––
\begin{thebibliography}{1}

\bibitem{priestley} Davey, Brian A., and Hilary A. Priestley. {\it Introduction to Lattices and Order}. Cambridge university press, 2002.

\bibitem{gratzer} Grätzer, George. {\it Lattice Theory: Foundation}. Springer Science $\&$ Business Media, 2011.

\bibitem{microarticles} Lobo, Matheus P. “Microarticles.” {\it OSF Preprints}, 28 Oct. 2019.\\\url{https://doi.org/10.31219/osf.io/ejrct}

\bibitem{ojmp-guidelines} Lobo, Matheus P. “Simple Guidelines for Authors: Open Journal of Mathematics and Physics.” {\it OSF Preprints}, 15 Nov. 2019.\\\url{https://doi.org/10.31219/osf.io/fk836}

\bibitem{ojmp-project} Lobo, Matheus P. “Open Journal of Mathematics and Physics (OJMP).” {\it OSF}, 21 Apr. 2020. \url{https://osf.io/6hzyp/files}

\bibitem{zen} \url{https://zenodo.org/record/7267788}

\bibitem{cc4} CC. Creative Commons. {\it CC-By Attribution 4.0 International}.\\\url{https://creativecommons.org/licenses/by/4.0}

\end{thebibliography}

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\section*{The Open Mathematics Collaboration}

\noindent {\bf Matheus Pereira Lobo}\textsuperscript{1} (\texttt{lead author,} \href{mailto:mplobo@uft.edu.br}{\texttt{mplobo@uft.edu.br}})\\ \url{https://orcid.org/0000-0003-4554-1372}\\

\noindent\textsuperscript{1}Federal University of Northern Tocantins (Brazil)

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\end{document}
