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

TL;DR

This work advances the finite-lattice enumeration problem by focusing on RC-lattices, where all reducible elements are comparable, and by leveraging a dismantlable, crown-free structure. It develops an adjunct-of-chains decomposition and a maximal-block framework, then provides a detailed, combinatorial method to enumerate basic blocks of nullity $k$ with $r$ comparable reducible elements via the class $B_r(k)$. By aggregating these blocks, the authors derive explicit counting formulas for maximal blocks and RC-lattices on $n$ elements, culminating in the main result $|\mathscr L(n)| = 1 + \sum_{k=1}^{n-3} |\mathscr L(n,k)|$ and related intermediate counts. The approach not only aligns with Birkhoff’s open problem on enumerating finite lattices but also yields practical enumeration pathways and two open questions for cases with $r \ge 4$ reducible elements.

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.

Theorems & Definitions (27)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• proof
• Definition 3.1