2017/11/16 by Gábor Czédli, Czédli, Gábor, Claudia Mureșan +1
Computer Science · #06B10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.1711.06394
openalex publication_date 2017/11/16 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
Let \λ and \κ be cardinal numbers such that \κ is infinite\nand either 2\≤ \λ\≤ \κ, or \λ=2^\κ. We prove that\nthere exists a lattice L with exactly \λ many congruences, 2^\κ\nmany ideals, but only \κ many filters. Furthermore, if \λ\≥ 2 is\nan integer of the form 2m\⋅ 3n, then we can choose L to be a modular\nlattice generating one of the minimal modular nondistributive congruence\nvarieties described by Ralph Freese in 1976, and this L is even relatively\ncomplemented for \λ=2. Related to some earlier results of George\nGr "atzer and the first author, we also prove that if P is a bounded ordered\nset (in other words, a bounded poset) with at least two elements, G is a\ngroup, and \κ is an infinite cardinal such that \κ\≥ |P| and\n\κ\≥ |G|, then there exists a lattice L of cardinality \κ such\nthat (i) the principal congruences of L form an ordered set isomorphic to\nP, (ii) the automorphism group of L is isomorphic to G, (iii) L has\n2^\κ many ideals, but (iv) L has only \κ many filters.\n