Generating sets for catalytic and projective distributive lattices
type: Item
instance of: scholarly article
title: Generating sets for catalytic and projective distributive lattices
zbMATH Open document ID: 0806.06008
DOI: 10.1007/BF01196096
author: Raymond Balbes
published in: Algebra Universalis
publication date: 1993-07-18T00:00:00Z
review text: A distributive lattice \(L\) is said to be catalytic if \(\text{Hom} (L,L_ 1)\) is a lattice under pointwise order for all distributive lattices \(L_ 1\). The author has already given a characterization of catalytic lattices [Algebra Univers. 11, 334-340 (1980; Zbl 0457.06008)]: Let \(L\) be a distributive lattice. Then \(L\) is catalytic if and only if \(L\) and \(L^ d\), the dual lattice of \(L\), satisfy the following two conditions: (1) Every element of them is a join of finitely many join-irreducible elements; (2) The meet of two join-irreducible elements is again join- irreducible. The author provides another characterization of the catalytic lattices: Let \(L\) be a distributive lattice satisfying the conditions (1) and (2). Then \(L\) is catalytic if and only if the meet-semilattice \(J(L)\) of join- irreducible elements is generated and dually generated by the set \(I(J(L))\) of all irreducible elements of \(J(L)\). It is well known that the projective distributive lattices are catalytic. The converse is true for the countable case. In addition, the author shows that the uncountable projective distributive lattices satisfy some antichain conditions.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B25
zbMATH DE Number: 203153
zbMATH Keywords: projective lattice
zbMATH Keywords: meet irreducible element
zbMATH Keywords: distributive lattice
zbMATH Keywords: catalytic lattices
zbMATH Keywords: dual lattice
zbMATH Keywords: join-irreducible elements
zbMATH Keywords: antichain
reviewed by: Tibor Katriňák
MaRDI profile type: Publication
cites work: Catalytic distributive lattices
cites work: Order sums of distributive lattices
cites work: Projective distributive lattices
cites work: Projective covers of distributive lattices
cites work: Q3728044
cites work: Catalytic distributive lattices and compact zero-dimensional topological lattices
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description: scientific article; zbMATH DE number 203153