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

Abstract

Let $λ$ and $κ$ be cardinal numbers such that $κ$ is infinite and either $2\leq λ\leq κ$, or $λ=2^κ$. We prove that there exists a lattice $L$ with exactly $λ$ many congruences, $2^κ$ many ideals, but only $κ$ many filters. Furthermore, if $λ\geq 2$ 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 $λ=2$. Related to some earlier results of George Grätzer 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 $κ$ is an infinite cardinal such that $κ\geq |P|$ and $κ\geq |G|$, then there exists a lattice $L$ of cardinality $κ$ 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 $2^κ$ many ideals, but (iv) $L$ has only $κ$ many filters.