Regular Double Heyting Algebras

• Regular Double Heyting Algebras are centrally supplemented Heyting algebras enhanced via hyper‐MacNeille completions and Boolean product representations.
• They integrate dual pseudo-complements and supplements to form free, universally characterized extensions across finitely generated varieties.
• Their structure underpins algebraic proof theory by facilitating categorical semantics and cut-elimination for intermediate and substructural logics.

A regular double Heyting algebra (abbreviated as RDBLH; Editor's term) emerges in the study of algebraic proof theory and the structural properties of Heyting algebras and their completions. The notion connects centrally supplemented Heyting algebras, dual pseudo-complements (supplements), and advanced completion methods, particularly the hyper-MacNeille completion, which is deeply tied to proof-theoretic cut-elimination for intermediate and substructural logics. The centrally supplemented Heyting algebras form the categorical framework underpinning regular double Heyting structures, with Boolean product representations and essential closure properties within finitely generated varieties (Harding et al., 2019).

1. Heyting Algebras, Supplements, and Center

A Heyting algebra is a bounded distributive lattice $(H, \land, \lor, \to, 0, 1)$, where the residual operation $\to$ satisfies the law

$a \land b \leq c \iff a \leq (b \to c).$

Pseudo-complements and supplements are dual notions in this context: The pseudo-complement (often $a^*$) is the greatest $x$ with $a \land x = 0$, given explicitly in a Heyting algebra by $a^* = a \to 0$. Dually, the supplement or dual pseudo-complement $a^+$ is the least $x$ such that $a \lor x = 1$, with

$\to$0

A supplemented lattice is one in which every element has a supplement; a centrally supplemented lattice requires every supplement to lie in the center $\to$1, the set of complemented elements. If $\to$2 is centrally supplemented, then its MacNeille completion $\to$3 is also centrally supplemented and $\to$4 preserves supplements (Harding et al., 2019).

2. Hyper-MacNeille Completions

The hyper-MacNeille completion is a categorical closure operation on Heyting algebras. For a Heyting algebra $\to$5, let $\to$6 with the relation

$\to$7

The associated Galois connection on $\to$8 yields a complete lattice $\to$9, termed the hyper-MacNeille completion and denoted $a \land b \leq c \iff a \leq (b \to c).$0. The key theorem establishes that $a \land b \leq c \iff a \leq (b \to c).$1 is isomorphic to the MacNeille completion of the centrally supplemented extension $a \land b \leq c \iff a \leq (b \to c).$2:

$a \land b \leq c \iff a \leq (b \to c).$3

The order-dense embedding $a \land b \leq c \iff a \leq (b \to c).$4 defined by $a \land b \leq c \iff a \leq (b \to c).$5, with $a \land b \leq c \iff a \leq (b \to c).$6, realizes the isomorphism (Harding et al., 2019).

3. Construction and Universal Property of Centrally Supplemented Extensions

Given any Heyting algebra $a \land b \leq c \iff a \leq (b \to c).$7, the centrally supplemented extension $a \land b \leq c \iff a \leq (b \to c).$8 is constructed as follows:

• Consider the Esakia space $a \land b \leq c \iff a \leq (b \to c).$9 of $a^*$0 and the set $a^*$1 of its minimal points (prime filters minimal under inclusion).
• For each $a^*$2, form the quotient $a^*$3, which is finite-subdirectly-irreducible and thus centrally supplemented.
• The product $a^*$4 is centrally supplemented.
• $a^*$5 is the smallest supplemented subalgebra of $a^*$6 containing the image of $a^*$7.

This extension $a^*$8 is centrally supplemented, lies in the same variety as $a^*$9, and enjoys a universal property: for any S-homomorphism $x$0 into a centrally supplemented Heyting algebra $x$1, there is a unique supplemented-homomorphism $x$2 extending $x$3. Thus, $x$4 is the free centrally supplemented extension (Harding et al., 2019).

4. Closure Properties and Boolean Product Representations

Finitely generated varieties of Heyting algebras are closed under hyper-MacNeille completion: If a variety $x$5 is generated by a finite algebra, then for any $x$6, $x$7. This closure follows from the structure of the minimal stalks $x$8, which are bounded in size, permitting sheaf-theoretic arguments.

Further,

• $x$9 admits a Boolean-product representation over the Stone space dual to $a \land x = 0$0:

$a \land x = 0$1

• The inclusion $a \land x = 0$2 is essential: every nontrivial congruence of $a \land x = 0$3 restricts nontrivially to $a \land x = 0$4.
• If $a \land x = 0$5 is centrally supplemented, then its subdirect decomposition over $a \land x = 0$6 gives $a \land x = 0$7, and $a \land x = 0$8 is the product of MacNeille completions of the finite quotients (Harding et al., 2019).

Object	Property	Construction/Significance
$a \land x = 0$9	Centrally supplemented	Generated as smallest such subalgebra of product of quotients
Hyper-MacNeille completion	Complete lattice	$a^* = a \to 0$0 for Galois connection on $a^* = a \to 0$1
Boolean-product representation	Internal structure of $a^* = a \to 0$2	Indexed by maximal central elements

5. Explicit Example: The Three-Element Chain

For $a^* = a \to 0$3:

• Esakia space $a^* = a \to 0$4 has two minimal points $a^* = a \to 0$5.
• Each quotient $a^* = a \to 0$6, the 2-element Boolean algebra.
• $a^* = a \to 0$7 is Boolean.
• The subalgebra generated by the diagonal embedding of $a^* = a \to 0$8 is all of $a^* = a \to 0$9, so $a^+$0.
• MacNeille completion is $a^+$1.
• Therefore, $a^+$2; under hyper-MacNeille, the chain completes to the 4-element Boolean square (Harding et al., 2019).

6. Significance in Algebraic Proof Theory

The hyper-MacNeille completion $a^+$3, interpreted as the MacNeille completion of $a^+$4, provides an algebraic route to the semantics of cut-free hypersequent calculi for broad classes of intermediate and substructural logics. This lattice-theoretic perspective offers an alternative to traditional frame-based or Kripke-style constructions. The existence and universal property of the centrally supplemented extension allow for robust categorical and universal algebraic treatments, facilitating deep connections between algebraic semantics and syntactic cut-elimination (Harding et al., 2019).

7. Connections and Broader Context

Regular double Heyting algebraic structures, as instantiated via centrally supplemented Heyting algebras and their completions, illuminate important closure properties for varieties, Boolean-product decompositions indexed by central elements, and connections to categorical universal properties. The algebraic semantics afforded by this approach directly address long-standing problems in algebraic proof theory and extend to important classes of logics, substantiating their role in the theory of residuated lattices and related algebraic systems (Harding et al., 2019).