On the size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras

Any finite distributive lattice has a unique Heyting structure. Every finite Heyting semilattice has, of course, a unique lattice structure and it is necessarily distributive. ∗ This ms was born in 2002. The first appendix was added in 2005, the first footnote in 2013, the subscorings in 2015 and the addendum in 2017. 1 [ ] See 2nd appendix for subscorings. First x and y meet x↔ y in the same way: x∧(x↔ y) = x∧((x∧x)↔ (x∧y)) = x∧(x↔ (x∧y)) = x∧((x∧1)↔ (x∧y)) = x∧(1↔ y) = x∧y and similarly y ∧ (x ↔ y) = x ∧ y. Hence for any z ≤ x ↔ y we have z ∧ x = z ∧ (x ↔ y) ∧ y = z ∧ x ∧ y and similarly z ∧ y = z ∧ (x ↔ y) ∧ y = z ∧ x ∧ y. Finally, if z ∧ x = z ∧ y then z ∧ (x ↔ y) = z ∧ ((z ∧ x) ↔ (z ∧ y)) = z ∧ 1 = z, that is, z ≤ x↔ y. Finitely generated hsls are finite.