ACTA issues

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

Gábor Czédli, Claudia Murełan

Acta Sci. Math. (Szeged) 85:3-4(2019), 363-380

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.

DOI: 10.14232/actasm-018-538-y

AMS Subject Classification (1991): 06B10

Keyword(s): lattice ideal, lattice filter, simple lattice, more ideals than filters, number of ideals, cardinality, lattice congruence, principal congruence

Received April 14, 2018 and in final form February 13, 2019. (Registered under 38/2018.)