# On distributive join-semilattices

## TL;DR

This paper introduces and explores a notion of distributivity in join-semilattices inspired by logical rules, establishing equivalences and extending to structures with implications relevant to intuitionistic logic.

## Contribution

It defines a new distributivity concept for join-semilattices, proves its equivalence to existing notions, and extends the concept to structures with arrow, linking algebraic and logical frameworks.

## Key findings

- Distributivity notions are linearly ordered.
- The new notion is equivalent to a known concept.
- Extension to join-semilattices with arrow connects to intuitionistic logic.

## Abstract

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.