A complete Heyting algebra is a complete lattice (H, ≤) equipped with binary meet (∧), binary join (∨), top (⊤), bottom (⊥), and an implication operation (⇒) satisfying the residuation law:
c ≤ (a ⇒ b) if and only if c ∧ a ≤ b
for all a, b, c ∈ H. Completeness means that meets and joins exist for all subsets, not just finite ones.
The residuation law defines the Heyting implication: a ⇒ b is the largest element whose meet with a stays below b. This gives H an internal logic — the elements of H serve as truth values, and the Heyting operations (∧, ∨, ⇒, ¬) provide the logical connectives. The logic is intuitionistic: the law of excluded middle (a ∨ ¬a = ⊤) may fail, which means H can represent partial or constructive truth.
Every complete Heyting algebra is equivalently a frame (a complete lattice where finite meets distribute over arbitrary joins) or a locale (the dual of a frame). This equivalence connects the algebraic structure to topology: the open sets of any topological space form a complete Heyting algebra, and complete Heyting algebras generalize topological spaces to settings where points may not exist.
In the semiotic universe, H is the ambient semantic domain. All semantic objects live in H, all logical reasoning uses H’s Heyting operations, and the modality j and trace comonad G are endomorphisms on H that respect its algebraic structure. The choice of a complete Heyting algebra (rather than a Boolean algebra) ensures that the semiotic logic is constructive — distinctions must be witnessed, not merely asserted by excluded middle.