Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements

TL;DR

This paper counts specific classes of lattices called RC-lattices with given nullity and reducible elements, advancing the enumeration of finite lattices and addressing Birkhoff's open problem.

Contribution

It introduces a comprehensive enumeration of RC-lattices with specified nullity and reducible elements, expanding the understanding of finite lattice structures.

Findings

Enumerated RC-lattices with nullity k and r reducible elements

Provided enumeration for RC-lattices on n elements with nullity k

Addressed Birkhoff's open problem on finite lattice enumeration

Abstract

In 2002 Thakare et al.\ counted non-isomorphic lattices on $n$ elements, having nullity up to two. In 2020 Bhavale and Waphare introduced the concept of RC-lattices as the class of all lattices in which all the reducible elements are comparable. In this paper, we enumerate all non-isomorphic RC-lattices on $n$ elements. For this purpose, firstly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$, and containing $2 \leq r \leq 2k$ reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on $n$ elements.