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

Abstract

Let $(L,\vee )$ be a finite n-element semilattice where $n\ge 5$. 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.