There are various ways to describe the Dedekind–MacNeille completion of a poset, the minimal complete lattice in which the poset can be embedded. I’ll first state the ones I’ve seen and then one I haven’t seen, which to me seems the most helpful and intuitive one.
I’ll use the same notation for all definitions to make it easier to compare them; I’ll write $A^\text u$ and $A^\text l$ for the sets of upper and lower bounds of $A$, respectively, in line with the Wikipedia article. The completion is always ordered by subset inclusion (in the case of cuts $(A,B)$, inclusion on $A$); I won’t repeat that in each definition.
MacNeille himself (in MacNeille, H. M. ($1937$). Partially ordered sets. Transactions of the American Mathematical Society, $42$($3$), $416$–$460$, available here) introduced the elements of the completion as cuts in analogy with Dedekind cuts of the rational numbers (Section $11$, p. $443$):
Definition $1$: The completion of a poset $S$ is the set of cuts $(A,B)$, where $A,B\subseteq S$, $A^\text u=B$ and $B^\text l=A$.
The Wikipedia article mentions this as an alternative definition; its main definition (which is also used in several answers on this site) is:
Definition $2$: The completion of a poset $S$ is the set of subsets $T\subseteq S$ such that $\left(T^\text u\right)^\text l=T$.
In his book Ordered Sets, Egbert Harzheim uses a related definition (Section $1.10$, p. $40$, available here):
Definition $3$: The completion of a poset $S$ is the set of subsets of $S$ of the form $\left(T^\text u\right)^\text l$, with $T\subseteq S$.
A definition which I believe is equivalent to these three, but which I haven’t seen anywhere, is:
Definition $4$: The completion of a poset $S$ is the set of subsets of $S$ of the form $T^\text l$, with $T\subseteq S$.
I’ll briefly show why these are all equivalent and then discuss their respective merits.
The operations $A\to A^\text u$ and $A\to A^\text l$ are anti-monotone: The more elements $A$ has, the fewer upper and lower bounds it has. The operations $A\to\left(A^\text u\right)^\text l$ and $A\to\left(A^\text l\right)^\text u$ are monotone (being the composition of two anti-monotone operations) and extensive (i.e. $A\subseteq\left(A^\text u\right)^\text l$ and $A\subseteq\left(A^\text l\right)^\text u$, since the elements of $A$ are lower bounds for the upper bounds of $A$ and vice versa).
Thus, $A^\text l\subseteq\left(\left(A^\text l\right)^\text u\right)^\text l$ (by extensivity of the outer pair of operations) and $A^\text l\supseteq\left(\left(A^\text l\right)^\text u\right)^\text l$ (by extensivity of the inner pair of operations and anti-monotonicity of the outer one), and so $A^\text l=\left(\left(A^\text l\right)^\text u\right)^\text l$. It follows that all sets of the form $T=A^\text l$ satisfy $\left(T^\text u\right)^\text l=T$. Conversely, all sets that satisfy $\left(T^\text u\right)^\text l=T$ are manifestly of the form $A^\text l$ (with $A=T^\text u$). This establishes the equivalence of Definitions $2$ and $4$. Also, the fact that any set of the form $A^\text l$ can be written as $\left(T^\text u\right)^\text l$ (with $T=A^\text l$), and conversely any set of the form $\left(T^\text u\right)^\text l$ is manifestly of the form $A^\text l$ (with $A=T^\text u$) establishes the equivalence of Definitions $3$ and $4$. The equivalence of Definitions $1$ and $2$ follows by elimination of $B$.
The advantages of Definition $1$ include that it maintains the duality between the sets $A$ and $B$ instead of arbitrarily choosing one of them and that it most clearly exhibits the analogy to Dedekind cuts of $\mathbb Q$.
An advantage of Definition $2$ over Definitions $3$ and $4$ is that it specifies a condition for a subset to be an element of the completion, whereas $3$ and $4$ specify a form, which generally yields many duplicates, e.g. $T^\text l$ will generally be the same for many different $T$.
An advantage of Definition $3$ over Definition $4$ is that the operation $A\to\left(A^\text u\right)^\text l$ is a closure operator, which can be useful.
An obvious advantage of Definition $4$ over Definition $3$ is that the form $T^\text l$ is simpler than the form $\left(T^\text u\right)^\text l$.
But what I really like about Definition $4$, and what makes me wonder why it’s never mentioned anywhere, is that to my mind it reflects much more clearly than the other definitions why this is in fact the minimal completion. We want every subset to have an infimum, and we want to introduce only as many new elements as necessary to achieve that. Introducing an element for each subset isn’t minimal because lots of subsets have the same set of lower bounds, and thus can be dealt with by introducing a single infimum for all of them. For each distinct set of lower bounds, we need an element to serve as its supremum, and thus as the infimum of all subsets that share this set of lower bounds. That’s exactly what Definition $4$ does.
(We need a separate element for each set of lower bounds because two subsets with different sets of lower bounds can’t have the same infimum, since the set of lower bounds is the initial segment determined by the infimum.)
I’m particularly puzzled why Harzheim uses Definition $3$ and never even mentions the possibility of the similar but simpler Definition $4$. Even if he prefers Definition $3$ because of the closure properties, it would have made sense to mention that the first step of forming a set of upper bounds is actually redundant and that the completion is simply the set of distinct sets of lower bounds.
My questions are:
- Is my proof that Definition $4$ is equivalent to the other three correct?
- If so, has this definition been mentioned or used anywhere?
- If it hasn’t, is there a disadvantage to it that I overlooked?