Aggregation on a Finite Lattice

Janowitz, M. F.; Radeleczki, Sándor

doi:10.1007/s11083-015-9373-9

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“…The atoms of V , denoted by At(V ) in the following, play an important role also in connection with tolerance simplicity. From [JanR15] we deduce (see also [JakPR16, 6.4]) the following: A finite atomistic lattice V satisfying T (0 V , a) = ∇ V for every atom a ∈ V , is tolerance simple.…”

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confidence: 99%

The lattice of congruence lattices of algebras on a finite set

Jakubíková-Studenovská

Pöschel

Radeleczki

2018

Algebra Univers.

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The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice E. We describe the atoms and coatoms. Each meet-irreducible element of E being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice E; in particular, we prove that E is tolerance-simple whenever |A| ≥ 4. Jakubíková-Studenovská/Pöschel/Radeleczki, Lattice of congruence lattices 2The subject of the present paper is related to the finite representation problem in its concrete version. For a fixed finite set A we consider all possible congruence lattices of algebras with base set A. These congruence lattices (ordered by inclusion) form itself a lattice E and we are going to investigate this lattice. An important tool is our knowledge about the lattice L of all quasiorder lattices of algebras defined on the set A described in [JakPR16] (using some techniques developed previously in the papers [JakPR11] and [JakPR13]). These two lattices are strongly interrelated: there is a residual mapping from L to E. Therefore, in Section 3, we investigate on abstract level, how lattice properties (which are relevant for us) behave under residual mappings (for instance, the coatoms of E directly can be obtained from the coatoms of L, see 3.1(iv)).Based on preliminary results from Section 2 and the results of [JakPR16] and Section 3, we describe the atoms (∨-irreducible elements), coatoms (Section 4) and further ∧-irreducible elements (Sections 5 and 6) of the lattice E. Finally, in Section 7, we investigate several lattice theoretic properties of E, e.g., it is tolerance simple, but has no properties related with modularity.

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confidence: 99%