Why order sturcture is important? (1)

In recent observations, it has become clear that the significance of the order structure is often overlooked in educational settings and textbooks. Recognizing this gap, the motivation behind this blog post is to shed light on the critical role that order structure plays in various mathematical domains.

The interplay between pre-order sets and numerous fields such as number theory, logic, category theory, and topology is well-documented in numerous essays on my blog. However, the intricacies of these connections might not initially pique everyone's interest.

Nevertheless, understanding the order structure becomes indispensable when delving into fundamental mathematical concepts such as topology, measure theory, groups, rings, and modules.

In this post, I will elucidate the importance of order structure and its ubiquitous presence in mathematics.

The Categories of order we will use are orders as follows.

$$\mathrm{Bool}\subset \mathrm{Lattice}\subset \mathrm{Pos}\subset \mathrm{Pre}$$

Functor to Pos

Topology

The definition of continuous and measurable functions can be traced back to a specific functor targeting the category of posets. Let us deal with the continuous function first.

Continuous function

The functor $F:{\mathrm{Top}}^{\mathrm{op}}\to \mathrm{Lattice}$$F:\mathrm{Top^{op}\to Lattice}$ is defined by:

$$(f:(X,{\tau }_X)\to (Y,{\tau }_Y))\mapsto (f^{-1}:{\tau }_Y\to {\tau }_X)$$

A function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$$f:(X,\tau_X)\to(Y,\tau_Y)$ is continuous if and only if $F(f)\in {\mathrm{Hom}}_{\mathrm{Lattice}}({\tau }_Y,{\tau }_X)$$F(f)\in\mathrm{Hom}_{\mathrm{Lattice}}(\tau_Y,\tau_X)$.

Open map

It is well known that every pre-order set $P$$P$ forms a Category, the objects are elements of $P$$P$, the morphisms are $⪯$$\preceq$.

For every $f:X\to Y$$f:X\to Y$, we can have two power set functors.

$P:\mathrm{Set}\to \mathrm{Set}$$P:\mathrm{Set}\to\mathrm{Set}$:

$$(g:X\to Y)\mapsto (g:P(X)\to P(Y))$$

and $\mathcal{P}:\mathrm{Set}\to \mathrm{Bool}$$\mathcal{P}:\mathrm{Set}\to\mathrm{Bool}$

$$(g:X\to Y)\mapsto (g^{-1}:(P(Y),\mathrm{\Delta },\cap )\to (P(X),\mathrm{\Delta },\cap ))$$

Both $g$$g$ and $g^{-1}$$g^{-1}$ are order-preserving maps, hence form two functors.

Indeed, there are a pair of adjoints.

$$g(A)\subseteq B⟺A\subseteq g^{-1}(B)$$

For a continuous function $f$$f$, we can view $f^{-1}$$f^{-1}$ as a functor from ${\tau }_Y$$\tau_Y$ to ${\tau }_X$$\tau_X$.

Notice that ${\tau }_X,{\tau }_Y$$\tau_X,\tau_Y$ are subcategories of $P(X),P(Y)$$P(X),P(Y)$.

Then for a continuous function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$$f:(X,\tau_X)\to(Y,\tau_Y)$, $f$$f$ is an open map if and only if $f$$f$ is the left adjoint of $f^{-1}$$f^{-1}$.

Since we already know that $f(A)\subseteq B⟺A\subseteq f^{-1}(B)$$f(A)\subseteq B \iff A\subseteq f^{-1}(B)$, the requirement $f⊢f^{-1}$$f \vdash f^{-1}$ is nothing but $f:{\tau }_Y\to {\tau }_X$$f:\tau_Y\to\tau_X$.

Measure

Measurable function

The functor $G:\mathrm{Measure}\to \mathrm{Bool}$$G:\mathrm{Measure}\to \mathrm{Bool}$ is defined by:

$$(f:(X,\mathrm{\Omega },\mu )\to (Y,\mathrm{\Sigma },\eta ))\mapsto (f^{-1}:\mathrm{\Sigma }\to \mathrm{\Omega })$$

A function $f:(X,\mathrm{\Omega },\mu )\to (Y,\mathrm{\Sigma },\eta )$$f:(X,\Omega,\mu)\to(Y,\Sigma,\eta)$ is measurable if and only if $G(f)\in {\mathrm{Hom}}_{\mathrm{Bool}}(\mathrm{\Sigma },\mathrm{\Omega })$$G(f)\in\mathrm{Hom}_\mathrm{Bool}(\Sigma,\Omega)$.

You could define something similar to an open map via the left adjoint of $f^{-1}$$f^{-1}$ existing.

Subobject and Quotient object

Definition. Let $\mathcal{C}$$\mathcal{C}$ be a category, and $X$$X$ be an object in $\mathcal{C}$$\mathcal{C}$. $(S,s)$$(S,s)$ is a subobject of $X$$X$ if $s:S\hookrightarrow X$$s:S\hookrightarrow X$ is a monomorphism.

Remark. $(X,{\mathrm{id}}_X)$$(X,\mathrm{id}_X)$ is a subobject of $X$$X$.

Examples include field extension, subset, subspace topology, subgroup, submodule...

Order structure of subject.

Definition. If $(R,r)$$(R,r)$ and $(S,s)$$(S,s)$ are both subobjects of $X$$X$, then we say that $(R,r)$$(R,r)$ is included in $(S,s)$$(S,s)$, or in symbol, $(R,r)⪯(S,s)$$(R,r)\preceq(S,s)$ if and only if $r$$r$ factors through $s$$s$. i.e., there is an arrow $h:R\to S$$h:R\to S$ such that $r=s\circ h$$r=s\circ h$.

Remark. $h$$h$ is automatically mono! Suppose $h\circ j=h\circ k$$h\circ j=h\circ k$, we need to show that $j=k$$j=k$.

$$s\circ (h\circ j)=(s\circ h)\circ j=r\circ j$$

Similarly,

$$s\circ (h\circ k)=(s\circ h)\circ k=r\circ k$$

and $h\circ j=h\circ k⟹r\circ j=r\circ k$$h\circ j= h\circ k\implies r\circ j=r\circ k$. By $r$$r$ being a monomorphism, $j=k.◻$$j=k.\quad\square$

In fact, here we get a subcategory of the slice category $\mathcal{C}/X$$\mathcal{C}/X$. We denote it as $(\mathrm{Sub}(X),⪯)$$(\mathrm{Sub}(X),\preceq)$, which is also a pre-order set.

We can quotient the isomorphism to get a partial order set $(\mathrm{Sub}(X),\le )$$(\mathrm{Sub}(X),\le)$.

Usually, we choose $(S,i)$$(S,i)$ to be the subset and the inclusion map to be the representative element when we work on a concrete category such as $\mathrm{Grp},\mathrm{Ring},\mathrm{Mod}(R)...$$\mathrm{Grp, Ring, Mod}(R)...$

For example, the subspace lattice of $V$$V$, where $V$$V$ is a vector space.

Definition. Let $\mathcal{C}$$\mathcal{C}$ be a category, and $X$$X$ be an object in $\mathcal{C}$$\mathcal{C}$. $(Q,q)$$(Q,q)$ is a quotient object of $X$$X$ if $q:X\twoheadrightarrow Q$$q:X\twoheadrightarrow Q$ is an epimorphism.

By duality, the quotient object is a subobject in ${\mathcal{C}}^{\mathrm{op}}$$\mathcal{C}^{\mathrm{op}}$. We denote the partial order set we could get as $(\mathrm{Quot}(X),\le )$$(\mathrm{Quot}(X),\le)$​.

Remark $\mathbb{Q}\in \mathrm{Quot}(\mathbb{Z})$$\Q\in\mathrm{Quot}(\Z)$, since $\iota :\mathbb{Z}\to \mathbb{Q}$$\iota:\mathbb Z\to\mathbb Q$ is a epimorphism!

Functor from $\mathcal{C}$$\mathcal{C}$ to Pos

Let $\mathcal{C}$$\mathcal{C}$ be a concrete category, the morphisms are structure-preserving maps.

We will deal with $\mathrm{Grp},\mathrm{Ring},\mathrm{Mod}(R)$$\mathrm{Grp, Ring, Mod}(R)$... in the rest of this essay.

We can construct a functor from $F:\mathcal{C}\to \mathrm{Pos}$$F:\mathcal{C}\to\mathrm{Pos}$ as follows.

$$F(X):=(\mathrm{Sub}(X),\subseteq )$$

The morphism becomes an order-preserving map between subobjects.

$$F(f):(\mathrm{Sub}(X),\subseteq )\to (\mathrm{Sub}(Y),\subseteq )$$

$F(f)$$F(f)$ has a fixed point, that is, $0$$0$.

Application of $F$$F$

Math Essays: A Functor from Grp to Pos and its application to Schur's Lemma in module theory.

Also, the proof of the proposition: If $|G|=p$$|G|=p$, then $G$$G$ is a cyclic group also depends on this functor.

Since $|G|=p$$|G|=p$ where $p$$p$ is a prime number implies $F(G)\cong P\{\ast \}$$F(G)\cong P\{*\}$. Hence pick $g\ne e$$g\ne e$, and the image of the group homomorphism $f:\mathbb{Z}\to G,g\mapsto g^n$$f:\mathbb{Z}\to G,g\mapsto g^n$ is $G$$G$.

By the way, the forgetful functor $U:\mathrm{Grp}\to \mathrm{Set}$$U:\mathrm{Grp}\to\mathrm{Set}$ is representable.

$$U(G)\cong {\mathrm{Hom}}_{\mathrm{Grp}}(\mathbb{Z},G)$$

Remark. For an abelian group $A$$A$, We can define $s:{\mathbb{Z}}^n\to A$$s:\mathbb{Z}^n\to A$ by considering $a_1,...,a_n\in A$$a_1,...,a_n\in A$ and

$$(m_1,...,m_n)\mapsto m_1{a}_1+...+m_n{a}_n$$

That is the span.