# Distributive laws in residuated binars

## TL;DR

This paper investigates the distributivity identities in residuated binars, revealing dependencies among certain identities and providing counterexamples to show the independence of others.

## Contribution

It establishes specific logical dependencies among distributivity identities in residuated binars with distributive lattice reducts and demonstrates their independence through counterexamples.

## Key findings

- Six pairs of identities imply another identity
- Counterexamples show no other dependencies exist
- Dependencies are specific to residuated binars with distributive lattices

## Abstract

In residuated binars there are six non-obvious distributivity identities of $\cdot$,$/$,$\backslash$ over $\wedge, \vee$. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.