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On principal congruences and the number of congruences of a lattice with more ideals than filters. (English) Zbl 1449.06005

The authors establish the following two main results:
Let \(\lambda\) and \(\kappa\) be cardinal numbers such that \(\kappa\) is infinite and either \(2\leq \lambda\leq \kappa\), or \(\lambda = 2^\kappa\). Then there exists a lattice \( L\) with exactly \(\lambda\) many congruences, \(2^\kappa\) many ideals, however only \(\kappa\) many filters. (Moreover if \(\lambda \geq 2\) is an integer of the form \(2^m\cdot 3^n\) then they can choose \(L\) to be a modular lattice generating one of the minimal modular non-distributive congruence varieties described by R. Freese [“Minimal modular congruence varieties”, Notices Am. Math. Soc. 23, #76T-A181 (1976)], and this \(L\) is even relatively complemented for \(\lambda = 2.\))
It is also proved that if \(P\) is a bounded poset with at least two elements, \(G\) is a group, and \(\kappa\) is an infinite cardinal such that \(\kappa \geq \vert P \vert\) and \(\kappa\geq \vert G \vert\), then there exists a lattice \(L\) of cardinality \(\kappa\) such that (i) the principal congruences of \(L\) yield 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.
The rather delicate and complex ways to verify all these statements are explained in detail in the paper.

MSC:

06B10 Lattice ideals, congruence relations
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