Analyzing Subuniverse Counts in Finite Semilattices: Unveiling the Rankings and Descriptions

TL;DR

This paper determines the specific counts of subuniverses in finite semilattices and characterizes the structures that achieve these counts, revealing the rankings of these counts among all such semilattices.

Contribution

It establishes exact values for the fourth, fifth, and sixth largest numbers of subuniverses in finite semilattices and describes the semilattices that attain these counts.

Findings

Fourth largest subuniverse count: 25·2^{n-5}

Fifth largest: 24.5·2^{n-5}

Sixth largest: 24·2^{n-5}

Abstract

Let $(L,\vee)$ be a finite n-element semilattice where $n\geq 5$. We prove that the fourth largest number of subuniverses of an $n$-element semilattice is $25\cdot 2^{n-5}$, the fifth largest number is $ 24.5\cdot 2^{n-5}$, and the sixth one is $ 24\cdot 2^{n-5}$. Also, we describe the $n$-element semilattices with exactly $25\cdot 2^{n-5}$, $ 24.5\cdot 2^{n-5}$ or $ 24\cdot 2^{n-5}$ subuniverses.