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

Let λ and κ be cardinal numbers such that κ is infinite and either 2 ≤ λ ≤ 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 ≥ 2 is an integer of the form 2 m · 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 ≤ | P | and κ ≤ | 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.