Counting of lattices containing up to $4$ reducible elements and having nullity up to $3$

TL;DR

This paper addresses Birkhoff's NP-complete problem by explicitly counting non-isomorphic lattices with specific properties, namely up to four reducible elements and nullity three, for finite sets.

Contribution

It provides a detailed enumeration of certain classes of finite lattices, contributing to the understanding of lattice structures and their classification.

Findings

Counted all non-isomorphic lattices with 4 reducible elements and nullity 3

Extended the enumeration to specific lattice classes within Birkhoff's problem

Contributed to the classification of finite lattices based on reducibility and nullity

Abstract

In this paper, we count all non-isomorphic lattices on $n$ elements, containing four reducible elements and having nullity three. This work is in respect of Birkhoff's open problem (which is NP-complete) of counting all finite lattices on $n$ elements.