On principal congruences and the number of congruences of a lattice with more ideals than filters

TL;DR

This paper constructs lattices with prescribed numbers of congruences, ideals, and filters, and explores their structural properties, including modularity and automorphism groups, extending previous results in lattice theory.

Contribution

It provides new constructions of lattices with specific congruence, ideal, and filter counts, including modular and relatively complemented lattices, and relates these to automorphism groups and principal congruences.

Findings

Existence of lattices with exactly λ congruences and 2^κ ideals but only κ filters.

Construction of modular lattices with prescribed principal congruences and automorphism groups.

Lattices with specified structural properties related to their ideals, filters, and congruences.

Abstract

Let $\lambda$ and $\kappa$ be cardinal numbers such that $\kappa$ is infinite and either $2\leq \lambda\leq \kappa$, or $\lambda=2^\kappa$. We prove that there exists a lattice $L$ with exactly $\lambda$ many congruences, $2^\kappa$ many ideals, but only $\kappa$ many filters. Furthermore, if $\lambda\geq 2$ is an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this $L$ is even relatively complemented for $\lambda=2$. Related to some earlier results of George Gr\"atzer and the first author, we also prove that if $P$ is a bounded ordered set (in other words, a bounded poset) with at least two elements, $G$ is a group, and $\kappa$ is an infinite cardinal such that $\kappa\geq |P|$ and $\kappa\geq |G|$, then there exists a lattice $L$ of cardinality $\kappa$ such that (i) the principal congruences of $L$ form an ordered set isomorphic to $P$, (ii) the automorphism group of $L$ is isomorphic to $G$, (iii) $L$ has $2^\kappa$ many ideals, but (iv) $L$ has only $\kappa$ many filters.