The structure of decomposable lattices determined by their prime ideals

TL;DR

This paper investigates the structure of decomposable lattices, a special class of distributive lattices, by analyzing their prime ideals and deriving properties for five specific types.

Contribution

It introduces the concept of decomposable lattices and characterizes their structure through prime ideals, including properties of five special subclasses.

Findings

Characterization of decomposable lattices via prime ideals

Properties of five special decomposable lattice classes

Structural insights into distributive lattices

Abstract

A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then 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 study the structure of decomposable lattices determined by their prime ideals. The properties for five special decomposable lattices are derived.