A lattice is a partial order in which every pair of elements has a meet (greatest lower bound) and a join (least upper bound).
The meet a ∧ b and join a ∨ b give the lattice an algebraic structure: two binary operations satisfying commutativity, associativity, idempotence, and the absorption laws (a ∧ (a ∨ b) = a and a ∨ (a ∧ b) = a). A lattice can equivalently be defined as an algebraic structure satisfying these laws, without reference to an underlying order — the order is then recoverable as a ≤ b if and only if a ∧ b = a.
A lattice is complete if meets and joins exist for all subsets, not just pairs. A lattice is bounded if it has a top element ⊤ (join of everything) and a bottom element ⊥ (meet of everything). A lattice is distributive if meet distributes over join: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).
A Heyting algebra is a bounded distributive lattice with an additional operation — the Heyting implication — satisfying the residuation law. A Boolean algebra is a Heyting algebra where every element has a complement (¬¬a = a). Every Boolean algebra is a Heyting algebra, but not conversely — Heyting algebras allow a richer, constructive logic.
The Heyting algebra H underlying the semiotic universe is a complete lattice: every collection of semantic values has a meet and a join. This completeness is what allows the three closure operators (semantic, syntactic, fusion) to have least fixed points.