The largest and all subsequent numbers of congruences of $n$-element lattices

Gábor Czédli

University of Szeged

Abstract: For a positive integer $n$, let SCL$(n)=\{|\textup{Con} (L)|: L$ is an $n$-element lattice$\}$ stand for the set of Sizes of the Congruence Lattices of $n$-element lattices. The $k$-th Largest Number of Congruences of $n$-element lattices, denoted by lnc $(n, k)$, is the $k$-th largest member of $\textup{SCL}\, (n)$. Let $(n_1,\dots,n_6):=(1,4,5,6,6,7)$, and let $n_k:=k$ for $k\geq 7$. In 1997, R. Freese proved that for $n\geq n_1=1$, $\textup{lnc}\, (n, 1)=2^{n-1}$. For $n\geq n_2$, the present author gave lnc $(n, 2)$. For $k=3,4,5$ and $n\geq n_k$, C. Mureşan and J. Kulin determined lnc $(n, k)$ in their 2020 paper. For $k\leq 5$ and $n\geq n_k$, the above-mentioned authors described the $n$-element lattices witnessing lnc $(n, k),$ too. For all positive integers $k$ and $n \geq n_k$, this paper determines $\textup{lnc} (n, k)$ and presents the lattices that witness it. It turns out that, for each fixed $k$, the quotient $\textup{lcd} (k):= \textup{lnc} (n, k)/ \textup{lnc} (n, 1)$ does not depend on $n\geq n_k$. Furthermore, lcd $(k)$ converges to $1/8$ as $k$ tends to infinity.

Keywords: Number of lattice congruences, Size of the congruence lattice of a finite lattice, Lattice with many congruences, Congruence density

Language: English

DOI: 10.15826/umj.2025.2.006

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