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 $n$.

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. Nevertheless, the elements of the exact pair serve together as a kind of minimal upper bound for the sequence. In this way, the exact pair phenomenon can be a weak replacement for the least-upper-bound property, in partial orders that are not complete in that sense, but which have exact pairs.

  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.

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?

I am interested, in part, in order to gain a greater understanding of how this property can be used, apart from the ways that I know from computability theory.

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...

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 $n$.

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. Nevertheless, the elements of the exact pair serve together as a kind of minimal upper bound for the sequence. In this way, the exact pair phenomenon can be a weak replacement for the least-upper-bound property, in partial orders that are not complete in that sense, but which have exact pairs.

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.

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?

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 $n$.

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. Nevertheless, the elements of the exact pair serve together as a kind of minimal upper bound for the sequence. In this way, the exact pair phenomenon can be a weak replacement for the least-upper-bound property, in partial orders that are not complete in that sense, but which have exact pairs.

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.

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?

I am interested, in part, in order to gain a greater understanding of how this property can be used, apart from the ways that I know from computability theory.