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.
AMS Subject Classification
more ideals than filters,
number of ideals,
Received April 14, 2018 and in final form February 13, 2019. (Registered under 38/2018.)