Catalytic distributive lattices and compact zero-dimensional topological lattices
type: Item
instance of: scholarly article
title: Catalytic distributive lattices and compact zero-dimensional topological lattices
zbMATH Open document ID: 0557.06006
DOI: 10.1007/BF01201099
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: Let D(0,1) denote the category of bounded distributive lattices and lattice homomorphisms preserving 0 and 1. A lattice L is said to be catalytic if Hom(L,M) is a lattice for all M. It is known from the work of \textit{R. Balbes} [Algebra Univers. 11, 334-340 (1980; Zbl 0457.06008)] that L is catalytic if and only if every element is a finite meet of elements in M(L) (the meet-irreducible elements), M(L) is a join subsemilattice, \(0\in M(L)\), and dual conditions hold for J(L), the join irreducibles. In this paper these conditions are translated to the Priestley dual in the category of compact totally order disconnected spaces. The principal result is that a lattice L is catalytic if and only if its dual is a topological lattice. This result is used to show L is catalytic if and only if it is a retract of a lattice freely generated by an ordered set.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 54G05
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 3887754
zbMATH Keywords: catalytic lattices
zbMATH Keywords: category of bounded distributive lattices
zbMATH Keywords: meet- irreducible elements
zbMATH Keywords: join irreducibles
zbMATH Keywords: Priestley dual
zbMATH Keywords: category of compact totally order disconnected spaces
zbMATH Keywords: retract
author: Hilary A. Priestley
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3887754