TITLE: Does the exact pair phenomenon for partial orders occur in your area of mathematics? 
QUESTION [31 upvotes]: 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...

REPLY [3 votes]: In trying to understand better a certain weak choice axiom, I once concocted an example in sheaf theory which involves an exact pair in the sense of this question, without knowing it was a "thing" elsewhere. 
In constructive mathematics, one choice principle which has been found useful is called the Presentation Axiom, Pax for short, which says roughly that any object $A$ (of one's given category) can be covered by an object $P$ which is projective: any further covering $Q \twoheadrightarrow P$ must split. You can imagine this applying for example to certain categories of sheaves of modules where one might want to consider projective resolutions. Or you can imagine it applying to certain categories of Heyting-valued sets or other toposes where the full axiom of choice is out of the question. 
In constructive mathematics, it is also known as "CoSHEP" (Category of Sets Has Enough Projectives); a more philosophical justification (quoting the nLab) is 

every set $A$ should have a ‘completely presented’ set of ‘canonical’ elements, that is elements given directly as they are constructed without regard for the equality relation imposed upon them. For canonical elements, equality is identity, so the BHK interpretation of logic justifies the axiom of choice for a completely presented set. This set is $P$, and $A$ is obtained from it as a quotient by the relation of equality on $A$. This argument can be made precise in many forms of type theory (including those of Martin-Löf and Thierry Coquand), which thus justify CoSHEP, much as they are widely known to justify dependent choice. 

Now, Pax as stated above (for let's say a topos $E$) was actually the external version: an object $P$ is externally projective if the external hom-functor $\hom(P, -): E \to Set$ takes epimorphisms to epimorphisms. There's also an internal version of Pax which is often more appropriate: if $(-)^P: E \to E$ denotes the internal hom-functor, then we say $P$ is internally projective if $(-)^P$ preserves epimorphisms, and internal Pax would say that every object $A$ is covered by an internally projective $P$. What I had wanted was an example of a topos which validates external Pax but not internal Pax. 
Presheaf toposes $E = Set^{C^{op}}$ (the category of functors $C^{op} \to Set$, where $C$ is a small category and $Set$ is just a vanilla category of sets in a ZFC background) always validate external Pax. It's easy to show that representable functors $\hom_C(-, a)$ are externally projective, as are coproducts of representable functors, and any object $F: C^{op} \to Set$ is covered by such a coproduct via a canonical map $\sum_{A \in Ob(C)} \sum_{x \in F A} hom_C(-, A) \twoheadrightarrow F$. So I wanted a $C$ which would force internal Pax to fail for $Set^{C^{op}}$. 
The example I came up with is where $C$ is a generic exact pair, i.e., the poset $\mathbb{N} \cup \{a, b\}$ where the natural numbers $\mathbb{N}$ is ordered as usual and $a, b$ are incomparable elements that dominate every element of $\mathbb{N}$. We consider $C$ as a 'thin' category as is customary for posets. I claim that $A = \hom_C(-, a)$, or indeed any object $P$ covering $A$, cannot be internally projective. Specifically, that such $(-)^P: E \to E$ cannot preserve an evident epimorphism $F \twoheadrightarrow G$ where $F(n) = \{m \in \mathbb{N}: m \geq n\}$ and $F(a), F(b)$ are empty, where the restriction maps are given by the inclusions $F(n+1) \hookrightarrow F(n)$, and where $G$ is the presheaf with $G(n) = \ast$ (a one-point set) and $G(a), G(b)$ are empty. 
To show the induced natural transformation $F^P \to G^P$ isn't an epimorphism, it suffices to show that the component $F^P(b) \to G^P(b)$ isn't a surjection between sets; in fact one may calculate that $F^P(b)$ is empty and $G^P(b)$ has exactly one element. Such calculations aren't exactly a spectator sport, so I'll wrap it up by making a few observations: 

For general $F$ one may identify $F^P(b)$ with $Nat(\hom_C(-, b) \times P, F)$, where the right side indicates a set of natural transformations. 
In the case where $P = \hom_C(-, a)$, we have in fact $\hom_C(-, b) \times \hom_C(-, a) \cong G$, and a natural transformation $G \to F$ boils down to having an element that belongs to the intersection of all the $F(n)$. Of course no such element exists since the intersection is empty. 
So $F^{\hom_C(-, a)}(b)$ is empty, and $G^{\hom_C(-, a)}(b)$ has an element (in fact just one). 
For general $P$ covering $\hom_C(-, a)$, there is a section $s: \hom_C(-, a) \to P$ (Yoneda lemma), and this section induces a map $F^P \to F^{\hom_C(-, a)}$, and thence a function $F^P(b) \to F^{\hom_C(-, a)}(b)$. The domain $F^P(b)$ has to be empty since the codomain is. On the other hand, $G^P(b)$ is nonempty because there is a map $\hom_C(-, b) \times P \to \hom_C(-, b) \times \hom_C(-, a) \cong G$.