Update algebraic_construction.tex

algebraic_construction.tex

@@ -217,28 +217,34 @@ \subsection{Relation}
         Equivalence & \poscell{Reflexive} & \poscell{Symmetric} & \poscell{Transitive} & \\ \bottomrule
     \end{tabular}
 \end{table}
-% \begin{table}[h]
-%     \centering
-%     \renewcommand{\arraystretch}{1.6}
-%     \begin{tabular}{l|l|l|l|l}
-%         \toprule
-%         \textbf{Homogeneous Relation}       & \multicolumn{1}{c|}{\textbf{Reflexivity}} & \multicolumn{1}{c|}{\textbf{Symmetry}} & \multicolumn{1}{c|}{\textbf{Transitivity}}     & \multicolumn{1}{c}{\textbf{Connectedness}}                 \\ 
-%         \midrule
-%         \midrule
-%         Directed graph & & & & \\ \midrule
-% Undirected graph & & Symmetric & & \\ \midrule
-% Dependency & Reflexive & Symmetric & & \\ \midrule
-% Tournament & Irreflexive & Asymmetric & & \\ \midrule
-% Preorder & Reflexive & & Transitive & \\ \midrule
-% Total preorder & Reflexive & & Transitive & Strongly Connected \\ \midrule
-% Partial order & Reflexive & Antisymmetric & Transitive & \\ \midrule
-% Strict partial order & Irreflexive & Asymmetric & Transitive & \\ \midrule
-% Total order & Reflexive & Antisymmetric & Transitive & Strongly Connected \\ \midrule
-% Strict total order & Irreflexive & Asymmetric & Transitive & Connected \\ \midrule
-% Partial equivalence & & Symmetric & Transitive & \\ \midrule 
-% Equivalence & Reflexive & Symmetric & Transitive & \\ \bottomrule
-%     \end{tabular}
-% \end{table}
+
+\dfn{Filtered Set}{
+    A \textbf{filtered set} or \textbf{directed set} is a preorder set $(X, \le)$ with an additional property that every pair of elements has an upper bound. In other words, for every $x, y \in X$, there exists $z \in X$ such that $x \le z$ and $y \le z$.
+}
+\prop{Intersection of equivalence relations is an equivalence relation}{
+    The intersection of a family of equivalence relations on a set $X$ is an equivalence relation on $X$.
+}
+\pf{
+    Suppose $(R_i)_{i\in i}$ is a family of equivalence relations on $X$. We can check that $\bigcap\limits_{i\in I}R_i$ is an equivalence relation on $X$ by checking the three properties of equivalence relation:
+    \begin{enumerate}[(i)]
+        \item Reflexivity: For any $x\in X$, since $(x,x)\in R_i$ for all $i\in I$, we have $(x,x)\in \bigcap\limits_{i\in I}R_i$.
+        \item Symmetry: For any $x,y\in X$,
+        \[
+            (x,y)\in \bigcap\limits_{i\in I}R_i\implies\forall i\in I,(x,y)\in R_i\implies\forall i\in I, (y,x)\in R_i\implies(y,x)\in \bigcap\limits_{i\in I}R_i
+        \]
+        \item Transitivity: For any $x,y,z\in X$,
+        \begin{align*}
+            (x,y)\in \bigcap\limits_{i\in I}R_i\text{ and }(y,z)\in \bigcap\limits_{i\in I}R_i
+            &\implies\forall i\in I,(x,y)\in R_i\text{ and }(y,z)\in R_i\\
+            &\implies\forall i\in I, (x,z)\in R_i\\
+            &\implies(x,z)\in \bigcap\limits_{i\in I}R_i
+        \end{align*}
+    \end{enumerate}
+}
+
+\dfn{Generated Equivalence Relation}{
+    Let $X$ be a set and $R\subseteq X\times X$ be a relation on $X$. The \textbf{generated relation} $\langle R \rangle$ is defined as the smallest equivalence relation on $X$ that contains $R$, or equivalently, the intersection of all equivalence relations on $X$ that contain $R$.
+}
 
 \section{Function}
 \prop{}{
@@ -679,6 +685,41 @@ \section{Limits and Colimits}
         \end{tikzcd}
     \]
 }
+
+\ex{Colimit of $\mathsf{Set}$-valued Functor}{
+    Let $F:\mathsf{J}\to \mathsf{Set}$ be a functor. Define a relation $\sim^*$ on $\coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$,
+    \[
+        (i,x)\sim^* (j,y)\iff \text{there exists }\lambda:i\to j\text{ in }\mathrm{Hom}_\mathsf{J}(i,j)\text{ such that }F(\lambda)(x)=y.
+    \]
+    Let $\sim$ denote the equivalence relation generated by $\sim^*$. Then 
+    \[
+        \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim
+    \]
+    and the map $\phi_i:F(i)\to \varinjlim F$ is given by the composition
+    \[
+        F(i)\xrightarrow{\quad\iota_i\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim
+    \]
+    If $\mathsf{J}$ is a filtered category, then the equivalence relation $\sim$ has a explicit description: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$,
+    \[
+        (i,x)\sim (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y).
+    \]
+}
+\pf{ 
+    It is easy to show 
+    \[
+        \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim
+    \]
+    by checking the universal property of $\varinjlim F$. If $\mathsf{J}$ is a filtered category, first we can check 
+    \[
+        (i,x)\approx (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y).
+    \]
+    is an equivalence relation. To show $\approx\;=\;\sim$, let's assume $\backsimeq$ is any equivalence relation containing $\sim^*$. If $(i,x)\approx(j,y)$, then there exists $\lambda_1:i\to k$ and $\lambda_2:j\to k$ such that $F(\lambda_1)(x)=F(\lambda_2)(y)=z$. Hence 
+    \[
+        (i,x)\sim^* (k,z)\text{ and }(j,y)\sim^* (k,z)\implies(i,x)\backsimeq (k,z)\text{ and }(j,y)\backsimeq (k,z)\implies (i,x)\backsimeq (j,y).
+    \]
+    This implies $\backsimeq$ contains $\approx$. Therefore, $\approx$ is the smallest equivalence relation containing $\sim^*$, which means $\approx$ coincides with $\sim$.
+}
+
 \dfn{Complete Category}{
     A category $\mathsf{C}$ is \textbf{complete} if it has all small limits. That is, for any functor $F:\mathsf{J}\to \mathsf{C}$ with $\mathsf{J}$ small, $\varprojlim F$ exists.
 }
@@ -744,6 +785,17 @@ \section{Limits and Colimits}
         (X,\leq)&\longmapsto \mathsf{X}
     \end{align*}
 }
+
+\ex{Filtered Set}{
+    A filtered set can be regarded as a filtered (0,1)-category with objects being elements of the set and morphisms being
+    \begin{align*}
+        \mathrm{Hom}(x,y)=\begin{cases}
+            \{*\} & \text{if }x\leq y\\
+            \varnothing & \text{otherwise}
+        \end{cases}
+    \end{align*}
+}
+
 \section{Representable Functor}
 \dfn{Presheaf}{
     Let $\mathsf{C}$ be a category. A \textbf{presheaf} on $\mathsf{C}$ is a functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$.
@@ -2396,7 +2448,7 @@ \subsection{Ideals}
 \dfn{Ideal generated from subset}{
     Let $R$ be a commutative ring and $\mathcal I(R)$ be the set of all ideals of $R$. Suppose $S\subseteq R$ be a subset. The \textbf{ideal generated by $S$}, denoted by $(S)$, is the smallest ideal of $R$ containing $S$, i.e. 
     \[
-        (S)=\bigcap_{\substack{ I\in \mathcal I(R)\\S\subseteq I}}I.
+        (S)=\bigcap_{\substack{ I\in \mathcal I(R)\\S\subseteq I}}I=\left\{\sum_{i=1}^n r_is_i\mid n\in\mathbb{Z}_{+},r_i\in R,s_i\in S\right\}.
     \]
     If $S=\{a_1,\dots,a_n\}$, we write 
     \[
@@ -2419,6 +2471,17 @@ \subsection{Ideals}
             \].
     \end{enumerate}
 }
+\prop{}{
+    Let $R$ be a commutative ring and $S$ be a subset of $R$. Then 
+    $$(S)=\sum_{s\in S}(s).$$
+}
+\pf{
+    \begin{align*}
+        \sum_{s \in S} (s)&=\left\{a_{s_1}+ \cdots +a_{s_n}\mid n\in\mathbb{Z}_{+},s_i\in S,a_{s_i}\in (s_i)\right\}\\
+        &=\left\{r_1s_{1}+ \cdots +r_ns_{n}\mid n\in\mathbb{Z}_{+},s_i\in S,r_i\in R\right\}\\
+        &=(S).
+    \end{align*}
+}
 
 \prop{Properties of Ideal Operations}{
     \begin{enumerate}[(i)]
@@ -2427,7 +2490,7 @@ \subsection{Ideals}
         \item ${I} ({J}+{K}) = {I}  {J}+{I}  {K}$
         \item $$
         \begin{gathered}
-        \left(\sum_{t \in T} I_t\right) J=\sum_{t \in T}\left(I_t J\right), \quad J\left(\sum_{t \in T} I_t\right)=\sum_{t \in T} J I_t.
+        J\sum_{t \in T} I_t=\sum_{t \in T} J I_t.
         \end{gathered}
         $$
         \item $I(J K)=(I J) K$
@@ -2442,6 +2505,21 @@ \subsection{Ideals}
     \end{enumerate}
 }
 
+\prop{}{
+    Let $I$ and $J$ be ideals of a commutative ring $R$ and $\mathfrak{p}$ be a prime ideal of $R$. Then 
+    \[
+        I\cap J\subseteq \mathfrak{p}\iff IJ\subseteq \mathfrak{p}\iff I\subseteq \mathfrak{p}\text{ or }J\subseteq \mathfrak{p}.
+    \]
+}
+\pf{
+    We have the following chain of implications:
+    \begin{itemize}
+        \item $I\cap J\subseteq \mathfrak{p}\implies IJ\subseteq \mathfrak{p}$. Note that $IJ\subseteq I\cap J$. The result follows immediately.
+        \item $IJ\subseteq \mathfrak{p}\implies I\subseteq \mathfrak{p}\text{ or }J\subseteq \mathfrak{p}$. Assume $IJ\subseteq \mathfrak{p}$. Suppose $I\subsetneq \mathfrak{p}$ and $J\subsetneq \mathfrak{p}$. Then there exist $a\in I-\mathfrak{p}$ and $b\in J-\mathfrak{p}$. Since $\mathfrak{p}$ is prime, $ab\in IJ\subseteq \mathfrak{p}$ implies $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$, which is a contradiction. Hence $I\subseteq \mathfrak{p}$ or $J\subseteq \mathfrak{p}$.
+        \item $I\subseteq \mathfrak{p}\text{ or }J\subseteq \mathfrak{p}\implies I\cap J\subseteq \mathfrak{p}$. Note that $I\cap J\subseteq I$. The result follows immediately.
+    \end{itemize}
+}
+
 \dfn{Radical Ideal}{
     An ideal $I$ is called a \textbf{radical ideal} if $I=\sqrt{I}$.
 }