Prime ideals in decomposable lattices

TL;DR

This paper studies the structure of prime, minimal prime, and special ideals within decomposable lattices, providing insights into their algebraic properties and how they differ from general distributive lattices.

Contribution

It introduces the concept of decomposable lattices and investigates their prime ideals, offering new understanding of their algebraic structure.

Findings

Characterization of prime ideals in decomposable lattices

Identification of minimal prime ideals and their properties

Analysis of special ideals and their role in the lattice structure

Abstract

A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b), b=\overline{b}\vee(a\wedge b)$ and $\overline{a}\wedge \overline{b}=0$. The main purpose of this paper is to investigate prime ideals, minimal prime ideals and special ideals of a decomposable lattice. These are keys to understand the algebraic structure of decomposable lattices.