  Motivated by Gentzen disjunction elimination rule in his Natural Deduction
calculus and reading inequalities with meet in a natural way, we conceive a
notion of distributivity for join-semilattices. We prove that it is equivalent
to a notion present in the literature. In the way, we prove that those notions
are linearly ordered. We finally consider the notion of distributivity in
join-semilattices with arrow, that is, the algebraic structure corresponding to
the disjunction-conditional fragment of intuitionistic logic.
