The join of two elements a and b in a partial order is their least upper bound: the smallest element c such that a ≤ c and b ≤ c.
Formally, a ∨ b = c requires: (1) a ≤ c and b ≤ c (c is an upper bound), and (2) for any d with a ≤ d and b ≤ d, c ≤ d (c is the least such). The join may not exist in an arbitrary poset; when it exists for every pair, the poset is a join-semilattice.
In a lattice, every pair has both a join and a meet. In a complete Heyting algebra, joins exist for arbitrary collections.
Join corresponds to disjunction (∨, “or”) in the internal logic: a ∨ b represents the weakest claim that entails both a and b. In Set, the join of two subsets is their union. In a topology, the join of any collection of open sets is their union (which is open by the arbitrary union axiom).
The modality j in the semiotic universe is join-continuous: j(⋁ᵢ aᵢ) = ⋁ᵢ j(aᵢ). This means the closure operator distributes over arbitrary disjunctions — stabilizing a join is the same as joining the stabilizations. This property ensures that the stable fragment H^st is a complete Heyting subalgebra.