When a quotient of a distributive lattice is a boolean algebra

TL;DR

This paper explores conditions under which a quotient of a distributive lattice, defined via a specific congruence related to an ideal and a derivation, results in a Boolean algebra.

Contribution

It introduces a new lattice congruence based on an ideal and a derivation, and characterizes when the quotient lattice is Boolean.

Findings

Identifies necessary and sufficient conditions for the quotient to be Boolean.

Defines a new congruence relation on distributive lattices.

Provides theoretical criteria for Boolean quotient structures.

Abstract

In this article, we introduce a lattice congruence with respect to a nonempty ideal $I$ of a distributive lattice $L$ and a derivation $d$ on $L$ denoted by $\theta_I^d$. We investigate some necessary and sufficient conditions for the quotient algebra $L/\theta_I^d$ to become a Boolean algebra.