 From an algebraic perspective, as we saw in  section 3,  there is a long tradition in natural language semantics that analyzes disjunction words as expressing a join operator in a Boolean algebra, which, at the sentential level, delivers the least upper bound of the two disjuncts with respect to classic entailment. Recent work showed that alternative-based systems don’t need to abandon the elegant uniform algebraic perspective of the classical analysis: while classical entailment gives rise to a Boolean algebra, inquisitive entailment gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators (Roelofsen 2013). Thus if we identify disjunction with the join operator in such a Heyting algebra, this automatically generates the desired cross-categorial, alternative-based notion (Ciardelli and Roelofsen 2015).