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

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.