Suppose that I have a partial order P and an increasing sequence $a_0< a_1<a_2<\cdots$ of elements of $P$. A pair of elements (b,c) from P is said to be an exact pair for this sequence, if
- Both $b$ and $c$ are upper bounds for the sequence, so that $a_n<b$ and $a_n<c$ for every $n$, and
- Whenever $d\leq b$ and $d\leq c$ then $d\leq a_n$ for some $n$.
b c
:
:
:
a_2
a_1
a_0
More generally, an ideal $I$ in $P$ admits an exact pair $(b,c)$, if $I=\{\ a \mid a< b\text{ and } a<c\ \}$.
Note that if a strictly increasing sequence has an exact pair, then the sequence cannot have a least upper bound, since such a bound would be below both b and c and therefore have to be below some $a_n$, and consequently not an upper bound of the sequence after all. Note also that if $b$ and $c$ form an exact pair for a strictly increasing sequence, then there can be no greatest lower bound for $\{b,c\}$, and so orders with such exact pairs are not lattices.
The exact pair property arises in computability theory, because in the hierarchy of Turing degrees, every increasing sequence admits an exact pair. This is one way of seeing that the set of Turing degrees is not a lattice. More generally, every countable ideal in the Turing degrees has an exact pair, and in the case of principal ideals, this implies that every Turing degree is the greatest lower bound of a pair of incomparable degrees. It also arises for certain hierarchies of complexity theory.
The exact pair property is so beautifully structural, serving as an alternative to completeness, and for this reason I have always wondered whether it could have applications in other contexts, but I have only ever heard of it in connection with the computability degrees. Therefore,
My question is: Does the exact pair phenomenon arise elsewhere in mathematics? Are there other natural orders studied outside logic that have the exact pair property?
Perhaps other natural hierarchies in mathematics exhibit the exact pair property? Or perhaps they do but this remains undiscovered…
Best Answer
The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds. This can be seen from the proof of Theorem 4.1 of my paper Filters in C*-algebras. Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero. As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice. To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$. Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see this mathoverflow post)