On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom

TL;DR

This paper explores the enumeration and structural relationships of certain lattice and closure system families, establishing bijections and providing algorithms, with specific counts for small base set sizes and conjectures for larger ones.

Contribution

It establishes one-to-one correspondences between Davis' set union lattices, atomic lattices, and closure systems, and provides enumerative algorithms and new structural insights.

Findings

Counted set closure systems for n=6

Established bijections via Galois adjunctions and Formal Concept Analysis

Provided algorithms and structural properties of atomic lattices

Abstract

In this paper we count set closure systems (also known as Moore families) for the case when all single element sets are closed. In particular, we give the numbers of such strict (empty set included) and non-strict families for the base set of size $n=6$. We also provide the number of such inequivalent Moore families with respect to all permutations of the base set up to $n=6$. The search in OEIS and existing literature revealed the coincidence of the found numbers with the entry for D.\ M.~Davis' set union lattice (\seqnum{A235604}, up to $n=5$) and $|\mathcal L_n|$, the number of atomic lattices on $n$ atoms, obtained by S.\ Mapes (up to $n=6$), respectively. Thus we study all those cases, establish one-to-one correspondences between them via Galois adjunctions and Formal Concept Analysis, and provide the reader with two of our enumerative algorithms as well as with the results of these algorithms used for additional tests. Other results include the largest size of intersection free families for $n=6$ plus our conjecture for $n=7$, an upper bound for the number of atomic lattices $\mathcal L_n$, and some structural properties of $\mathcal L_n$ based on the theory of extremal lattices.