nLab MacNeille completion

Idea

The MacNeille or Dedekind–MacNeille (not ‘Mac Neille’) completion of a lattice (or even poset) $L$ is the universal complete lattice containing $L$. In Pos, the MacNeille completion of an object is its injective hull.

Definition

A MacNeille completion of $L$ is any complete lattice $C$ containing $L$ as a sublattice, with $L$ both join-dense and meet-dense in $C$ (that is, each element of $C$ is both a join of elements of $L$ and a meet of elements of $L$), and such that if $C'$ is another such lattice then there is a unique isomorphism from $C$ to $C'$ fixing $L$. In particular, this allows us to speak of the MacNeille completion of $L$.

A MacNeille completion may be constructed as follows. Let $P(L)$ be the set of all subsets of $L$, ordered by inclusion. For any $X \in P(L)$ let

• $X^{\Delta} = \{\alpha \in L \mid \forall x \in X . \alpha \geq x \}$

• $X^{\nabla} = \{\alpha \in L \mid \forall x \in X . \alpha \leq x \}$

These form a Galois connection on $P(L)$; the condition $\phi(X) = (X^{\Delta})^{\nabla}$ defines a closure operation $\phi$ on $P(L)$. The lattice $C$ of all $\phi$-closed subsets of $L$ is complete. For any $x \in L$ the set $i(x) = (\{x\}^{\Delta})^{\nabla}$ is the principal ideal generated by $x$. Then $i$ is an isomorphic embedding of $L$ into the complete lattice $C$ that preserves all least upper bounds (joins) and greatest lower bounds (meets) existing in $L$.

Examples

When applied to the ordered set of rational numbers, the construction described above gives the completion of the set of rational numbers by Dedekind sections, including the empty sections. Thus we get, not the real line $\mathbb{R} = {]{-\infty},\infty[}$, but the extended real line $\overline{\mathbb{R}} = [-\infty,\infty]$. In constructive mathematics, we may get additional finite elements, since the Dedekind sections are not required to be located; if we then drop $\pm\infty$, we get the (bounded) MacNeille real numbers.

Related pages

• The MacNeille completion can be categorified to the category of self-dual (saturated) objects in the Isbell envelope; see this MO question and answers.

References

• MacNeille completions and canonical extensions
• Encyclopedia of Mathematics
• Section 21H of The Joy of Cats (web)