Mathematica Pannonica
Volume/Issue:
Volume 28_NS2: Issue 2
A Property of Lattices of Sublattices Closed Under Taking Relative Complements and Its Connection to 2-Distributivity
Author:
Gábor Czédli University of Szeged, Hungary

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Pages:
109–117
Online Publication Date:
26 May 2022
Publication Date:
29 Sep 2022
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2022.00014
Keywords:
Relative complement; modular lattice; 2-distributive lattice; n-distributivity; sublattice; ranked lattice; Jordan–Hölder chain condition
Open access

For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.

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Mathematica Pannonica
Print ISSN:
0865-2090
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2786-0752

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Mathematica Pannonica
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ISSN 2786-0752 (Online)
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