Hyper–MacNeille Completion

• Hyper–MacNeille Completion is a categorical lattice-theoretic construction that extends any Heyting algebra to a complete, centrally supplemented extension using Galois connections.
• It generalizes the classical MacNeille completion by applying the method to a centrally supplemented extension, thus preserving join and meet operations while addressing proof-theoretic needs.
• This method provides a uniform algebraic semantics for cut-free hypersequent calculi, deepening insights into the structure and variety of Heyting algebras.

The hyper–MacNeille completion is a categorical lattice-theoretic construction on Heyting algebras that associates to any Heyting algebra a specific complete extension, generalizing the classical MacNeille completion in a manner attuned to algebraic proof theory and dualities for non-classical logics. It is defined via Galois connections on the cartesian square of a Heyting algebra and is characterized, up to isomorphism, as the MacNeille completion of a canonical centrally supplemented extension of the original algebra. The construction has consequences for the structure theory of Heyting algebras, the closure properties of their varieties, and the algebraic underpinnings of hypersequent calculi.

1. Foundational Notions: Heyting Algebras and Standard MacNeille Completion

A Heyting algebra is a bounded distributive lattice $(H, \land, \lor, \to, 0, 1)$ where $\to$ is a binary operation satisfying the residuation property: $a\land b \leq c$ if and only if $a \leq (b\to c)$ for $a,b,c\in H$. The MacNeille completion $\overline{H}$ of $H$ is the smallest complete lattice into which $H$ embeds join- and meet-densely. This completion can be realized as the lattice of all Galois-closed subsets (cuts) of $H$ with respect to the order $\le$, making $\to$0 universal among complete lattice extensions of $\to$1 that preserve meets and joins (Harding et al., 2019).

2. Supplements, Pseudo-Complements, and Central Elements

For a bounded distributive lattice $\to$2, the pseudo-complement of $\to$3 is $\to$4, while the dual pseudo-complement (supplement) is $\to$5. A lattice is supplemented if every element has a supplement. Centrality of supplements is defined by $\to$6's being a complemented element; that is, $\to$7, the center of the lattice. The dual Stone law, $\to$8, and the inclusion of all supplements in $\to$9 are equivalent in this context. If every element's supplement is central, the algebra is centrally supplemented (Harding et al., 2019).

3. Construction of the Hyper–MacNeille Completion

Given a Heyting algebra $a\land b \leq c$0, consider $a\land b \leq c$1 and define a binary relation $a\land b \leq c$2 on $a\land b \leq c$3: $a\land b \leq c$4 Let $a\land b \leq c$5 and $a\land b \leq c$6 denote the Galois operators induced by $a\land b \leq c$7. The hyper–MacNeille completion $a\land b \leq c$8 is then the lattice of Galois-closed subsets of $a\land b \leq c$9: $a \leq (b\to c)$0. With pointwise lattice operations and suitably defined implication,

$a \leq (b\to c)$1

for $a \leq (b\to c)$2, $a \leq (b\to c)$3 forms a complete Heyting algebra extending $a \leq (b\to c)$4 (Harding et al., 2019).

4. Centrally Supplemented Extensions and the Main Theorem

To analyze $a \leq (b\to c)$5, construct the centrally supplemented extension $a \leq (b\to c)$6 of $a \leq (b\to c)$7:

• Let $a \leq (b\to c)$8 be the set of minimal prime filters of $a \leq (b\to c)$9.
• For each $a,b,c\in H$0, define the congruence

$a,b,c\in H$1

and set $a,b,c\in H$2 (which is finitely subdirectly irreducible and centrally supplemented).

• Form $a,b,c\in H$3, and let $a,b,c\in H$4 be the subalgebra generated by the diagonal copy of $a,b,c\in H$5.

$a,b,c\in H$6 is centrally supplemented, the embedding $a,b,c\in H$7 is subdirect and essential, and $a,b,c\in H$8 lies in the same variety as $a,b,c\in H$9. Importantly, any homomorphism $\overline{H}$0 into a centrally supplemented Heyting algebra $\overline{H}$1 extends uniquely to $\overline{H}$2. The main structural result is: $\overline{H}$3

where $\overline{H}$4 denotes the MacNeille completion of $\overline{H}$5, realizing $\overline{H}$6 as the "MacNeille completion of a centrally supplemented extension" (Harding et al., 2019).

5. Relationship with Varieties and Boolean-Product Representations

If $\overline{H}$7 is a variety of Heyting algebras generated by a finite Heyting algebra $\overline{H}$8, then any algebra $\overline{H}$9 in $H$0 has its centrally supplemented extension $H$1, and so $H$2 also lies in $H$3. Stalks of the sheaf representation of $H$4 carry at most $H$5 elements on a dense open set. The extension $H$6 admits a representation as a weak Boolean product of its factors $H$7, with indexing by the Stone space of central ultrafilters. Conversely, any Boolean-product representation of $H$8 with finitely subdirectly irreducible stalks recovers a centrally supplemented structure. Thus, the hyper–MacNeille procedure preserves finitary variety structure and is closely related to Boolean-product decompositions (Harding et al., 2019).

6. Explicit Example and Illustrative Case

For the three-element chain $H$9:

• The minimal prime filters are $H$0.
• The quotients $H$1 and $H$2.
• Thus, $H$3, and $H$4, generated by $H$5, equals the full product.
• $H$6 is centrally supplemented, and the MacNeille completion of $H$7 is itself, giving $H$8.
• The original $H$9 embeds diagonally, with computations in $H$0 agreeing with the Galois-closed-subset description (Harding et al., 2019).

7. Significance in Algebraic Proof Theory

The hyper–MacNeille completion provides a uniform algebraic semantics for cut-free hypersequent calculi in substructural logics. By embedding any Heyting algebra into a complete, centrally supplemented extension whose MacNeille completion matches the proof-theoretic frame, this method yields a purely algebraic account of cut-admissibility and related structural rules in non-classical logics. This suggests deeper categorical and algebraic connections between the semantic and syntactic structure of substructural logics and the intrinsic order-completion procedures on their algebraic semantics (Harding et al., 2019).