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

Let (L,) be a finite n-element semilattice where n≥ 5. We prove that the fourth largest number of subuniverses of an n-element semilattice is 25· 2ⁿ⁻⁵, the fifth largest number is 24.5· 2ⁿ⁻⁵, and the sixth one is 24· 2ⁿ⁻⁵. Also, we describe the n-element semilattices with exactly 25· 2ⁿ⁻⁵, 24.5· 2ⁿ⁻⁵ or 24· 2ⁿ⁻⁵ subuniverses.