The connection between three kind of glb and lub (union, intersection, max,min, and gcd, lcm)

Consider the characteristic function $\begin{array}{c}{\chi }_U(x):=\{\begin{array}{l}1,x\in U\\ 0,x\notin U\end{array}\end{array}$$\chi_U(x):=\begin{cases}1,x\in U\\0,x\notin U\end{cases}$

It can be generalized to multiset $\begin{array}{c}{\chi }_U(x):=\{\begin{array}{l}|x|,x\in U\\ 0,x\notin U\end{array}\end{array}$$\chi_U(x):=\begin{cases}|x|,x\in U\\0,x\notin U\end{cases}$

Where $|x|$$|x|$ means how many $x$$x$ in $U$$U$

For example, $U=\{1,1,1,2,2,3,3,3,3\}$$U=\{1,1,1,2,2,3,3,3,3\}$,${\chi }_U(1)=3,{\chi }_U(2)=2,{\chi }_U(3)=4$$\chi_U(1)=3,\chi_U(2)=2,\chi_U(3)=4$

Consider all the characteristic functions of $X$$X$, denoted as $\chi$$\chi$

Define a partial order $⪯$$\preceq$ over $\chi$$\chi$ :${\chi }_U⪯{\chi }_V\iff \mathrm{\forall }x,{\chi }_U(x)\le {\chi }_V(x)$$\chi_U\preceq\chi_V\Leftrightarrow \forall x,\chi_U(x)\le\chi_V(x)$

Easy to see that ${\chi }_U⪯{\chi }_V\iff U\subseteq V$$\chi_U\preceq\chi_V \Leftrightarrow U\subseteq V$.

It is a natural partial-order isomorphism, denoted as $I$$I$

Thus $Imax({\chi }_U,{\chi }_V)=U\cup V,Imin({\chi }_U,{\chi }_V)=U\cap V$$I\max(\chi_U,\chi_V)=U\cup V,I\min(\chi_U,\chi_V)=U\cap V$

Easy to see the connection between gcd, and lcm.

Math Essays: The connection between Boolean Algebra and Number Theory. (wuyulanliulongblog.blogspot.com)

And observe that all the functions to $\mathbb{N}$$\mathbb N$ can be viewed as characteristic functions of a multiset

Thus $gcd,\mathrm{l}\mathrm{c}\mathrm{m}$$\gcd,\rm lcm$, $\cap ,\cup ,min,max$$\cap,\cup,\min,\max$ is the same kind of glb and lub