The lattice of sets of natural numbers is rich

\newcommand\of{\subseteq}\newcommand\N{\mathbb{N}}\newcommand\R{\mathbb{R}}\def\<#1>{\left\langle#1\right\rangle}\newcommand\Z{\mathbb{Z}}\newcommand\Q{\mathbb{Q}}\newcommand\ltomega{{{<}\omega}}\newcommand\unaryminus{-}\newcommand\intersect{\cap}\newcommand\union{\cup}\renewcommand\emptyset{\varnothing}Let us explore the vast and densely populated expanses of the lattice of all sets of natural numbers—the power set lattice $\<P(\N),\of>$. We find in this lattice every possible set of natural numbers $A\of\N$, and we consider them with respect to the subset relation $A\of B$. This is a lattice order because the union of two sets $A\union B$ is their least upper bound with respect to inclusion and the intersection $A\intersect B$ is their greatest lower bound. Indeed, it is a distributive lattice in light of $A\intersect(B\union C)=(A\intersect B)\union(A\intersect C)$, but furthermore, as a power set algebra it is a Boolean algebra and every Boolean algebra is a distributive lattice. The empty set $\emptyset$ is the global least element at the bottom of the lattice, of course, and the whole set $\N$ is the greatest element, at the top. The singletons $\set{n}$ are atoms, one step up from $\emptyset$, while their complements $\N\setminus\set{n}$ are coatoms, one step down from $\N$. One generally conceives of the finite sets as clustering near the Earthly bottom of the lattice, finitely close to $\emptyset$, whereas the cofinite sets soar above, finitely close to the heavenly top. In the vast central regions between, in contrast, we find the infinite-coinfinite sets, including many deeply interesting sets such as the prime numbers, the squares, the square-free numbers, the composites, and uncountably many more. Question. Which familiar orders can we find as suborders in the power set lattice $P(\N)$ of all sets of natural numbers? We can easily find a copy of the natural-number order $\<\N,\leq>$ in the power set lattice $P(\N)$, simply by considering the chain of...