Exploring Multisets, Lattices, Number Theory, and Modules in Mathematical Essays

I have already written some blogs about multisets, lattices, number theory, and modules.

Math Essays: The Connection Between Boolean Algebra and Number Theory (wuyulanliulongblog.blogspot.com)

Math Essays: Coprime-Orthogonal, Module, and p-adic Valuation (wuyulanliulongblog.blogspot.com)

Math Essays: Lattice Over Continuous Functions (wuyulanliulongblog.blogspot.com)

Math Essays: The Connection Between Three Kinds of Greatest Lower Bound (glb) and Least Upper Bound (lub) (Union, Intersection, Max, Min, and gcd, lcm) (wuyulanliulongblog.blogspot.com)

Math Essays: Lattice and Boolean Algebra over vector space (1) (wuyulanliulongblog.blogspot.com)

Math Essays: Lattice and Boolean Algebra over vector space (2) ----- inclusion-exclusion theorem and measure space (wuyulanliulongblog.blogspot.com)

I have an initial idea about how to formally define a multiset. Today, I came across a formal definition that aligns with my thoughts!

https://ncatlab.org/nlab/show/multiset

Here's the formal definition:

Definition:

A multiset $M_X:=(X,{\mu }_X)$$M_X := (X, \mu_X)$, where $\mu :X\to \mathbb{N}$$\mu: X \to \mathbb{N}$.

Here, $\mu (x)$$\mu(x)$ for $x\in X$$x \in X$ tells us how many times $x$$x$ appears in $M_X$$M_X$.

• The cardinality of $M_X$$M_X$ is given by $|M_X|=\sum _{x\in X}\mu (x)$$|M_X| = \sum_{x \in X} \mu(x)$.

• The union of two multisets is given by $M_X\cup M_Y=(X\cup Y,max({\mu }_X,{\mu }_Y))$$M_X \cup M_Y = \big(X \cup Y, \max(\mu_X, \mu_Y)\big)$.

• The intersection of two multisets is given by $M_X\cap M_Y=(X\cap Y,min({\mu }_X,{\mu }_Y))$$M_X \cap M_Y = \big(X \cap Y, \min(\mu_X, \mu_Y)\big)$.

• The difference of two multisets is given by $M_X-M_Y=(a\in X\cup Y|{\mu }_X(a)\ge {\mu }_Y(a),{\mu }_X-{\mu }_Y)$$M_X - M_Y = \big({a \in X \cup Y | \mu_X(a) \ge \mu_Y(a)}, \mu_X - \mu_Y\big)$.

• The sum of two multisets is given by $M_X+M_Y=(X\cup Y,{\mu }_X+{\mu }_Y)$$M_X + M_Y = (X \cup Y, \mu_X + \mu_Y)$.

• The product of two multisets is given by $M_X{M}_Y=(X\cap Y,{\mu }_X{\mu }_Y)$$M_XM_Y = (X \cap Y, \mu_X \mu_Y)$.

• The inner product of multisets is given by $⟨M_X,M_Y⟩=\sum _{e\in X\cap Y}{\mu }_X(e){\mu }_Y(e)$$\langle M_X, M_Y\rangle = \sum_{e \in X \cap Y} \mu_X(e) \mu_Y(e)$.

This can be seen as an inner product. Consider a universal set $S$$S$ such that $X\cup Y\subseteq S$$X \cup Y \subseteq S$, and represent $\mu \mapsto (\mu (s_1),\mu (s_2),...,\mu (s_n))$$\mu \mapsto (\mu(s_1), \mu(s_2), ..., \mu(s_n))$.

It is easy to see that the inner product gives the cardinality of $M_X\cap M_Y$$M_X \cap M_Y$.

An interesting example is ${\mathbb{N}}^{+}$$\mathbb{N}^+$, where every natural number $n$$n$ can be viewed as $M_n:=(\mathbb{P},{\mu }_n)$$M_n := (\mathbb{P}, \mu_n)$, and ${\mu }_n(p)=v_p(n)$$\mu_n(p) = v_{p}(n)$.

This is essentially the fundamental theorem of arithmetic.

$M_n\cup M_m=gcd(n,m)$$M_n \cup M_m = \gcd(n, m)$, $M_n\cap M_m=\mathrm{lcm}(n,m)$$M_n \cap M_m = \mathrm{lcm}(n, m)$, $M_n+M_m=nm$$M_n + M_m = nm$.

Also, $M_n\cup M_m+M_n\cap M_m=M_n+M_m$$M_n \cup M_m + M_n \cap M_m = M_n + M_m$.

$gcd(n,m)=1⟺⟨M_n,M_m⟩=0$$\gcd(n, m) = 1 \iff \langle M_n, M_m\rangle = 0$, thus coprime means orthogonal.

Disjoint is orthogonal too.

For more examples and interesting discussions, you can click on the links above.