Properties of principal moore families

I previously introduced principal moore families, and I presented an algorithm to test for them. However, I didn't discuss any of their set-theoretic properties. In this post I will describe how principal moore families are families of principal ideals (or principal filters) of certain preorders with lattice condensation. In order to do this, we need to describe properties of unions and intersections of ideals.

Proposition. principal moore families are union-free

Proof. let $S$ and $T$ be two inclusion-incomparable sets of the moore family and suppose their union $S \cup T$ is contained in the moore family. Then by the definition of principal moore families there exists an element $x$ such that $cl(\{x\}) = S \cup T$. But the definition of the closure implies that the closure of a set is the smallest set containing it, and since $x$ is in either $S$ or $T$ and both of them are less then their union the closure must be $S$ or $T$ and not $S \cup T$. Therefore, by contradiction the family is union-free.

It is well known that subgroups, submodules, vector subspaces, subrings, ideals, subalgebras, and so on are all union-free. It should not come as a suprise then that the family of ideals in a PID is union-free and so on, but this theorem applies in general.

Question. when is the intersection of principal ideals of a preorder a principal ideal?

Answer. let $S$, $T$ be principal ideals and consider their intersection $cl(\{s\}) \cap cl(\{t\})$ in order for this intersection to be a principal ideal there must be some element $x$ for which the principal ideal $\{ y : y \subseteq x \}$ is equal to $\{ y : y \subseteq s \land y \subseteq t \}$. In other words, $y \subseteq x \iff y \subseteq s \land y \subseteq t$. The first inclusion means that $x$ must be a lower bound of $s$ and $t$ and the second inclusion which states that all elements are less then it means that $x$ must be the greatest lower bound. Therefore, intersections of principal ideals correspond to meets when they exist.

Theorem. principal moore families are preorder containment families of upper-bounded meet-complete prelattices

Proof. let $M$ be a moore family, and define the generation preorder to be $a \leq b$ is logically equivalent to $cl(\{a\}) \subseteq cl(\{b\})$ then $M$ is the family of principal ideals of this preorder. To see this note that $cl({a}) \subseteq cl(\{b\})$ logically implies that $\{a\} \subseteq cl({a}) \subseteq cl({b})$ because the closure operation is extensive, which means that $a \in cl({b})$. Suppose that we have an element $x$ that isn't a predecessor but which is contained in $cl({b})$ then this means that the closure $cl(\{x\})$ is not contained in $cl(\{b\})$ but this contradicts the definition of closure as the smallest parent set. It follows that $M$ is a preorder containment family.

Moore families are upper-bounded meet-complete lattices, so it further follows that this preorder has an upper bounded meet-complete lattice as its condensation. In the other direction, in order for the family of all principal ideals to be a Moore family, it must have all intersections, but these correspond to meets so intersection closure follows from meet completeness. The upper bound is provided by supposition, so the family of principal ideals forms a principal moore family. Therefore, the two concepts are equivalent.

Corollary. the family of principal ideals of a finite lattice is a principal kolmogorov moore family

The two different ways of presenting a preorder are either as preorder containment families or alexandrov families. The only real difference between the two is that the first is union-free and the later is union-closed. If we take a finite lattice, and get the union closure of its principal kolmogorov moore family then we will get a kolmogorov alexandrov topology whose inclusion order corresponds to a distributive lattice with the original lattice as its suborder of join irreducibles.

In the case of PIDs, it suffices to consider the ideal generation preorder on elements. This preorder is clearly a meet-complete upper bounded prelattice, with the equivalence class of units as its preorder upper bound. The resulting family of ideals is entirely determined by the resulting preorder and individual ideals are its symmetric components. The lattice of ideals is determined by the condensation.