Doubling convex sets in lattices and a generalized semidistributivity condition
type: Item
instance of: scholarly article
title: Doubling convex sets in lattices and a generalized semidistributivity condition
zbMATH Open document ID: 0695.06005
DOI: 10.1007/BF02034334
author: Alan Day
author: J. B. Nation
author: S. Tschantz
published in: Order
publication date: 1989-01-01T00:00:00Z
review text: A useful construction for lattices is doubling a convex subset I of a lattice L, i.e., replacing I by \(I\times 2\). In case I is an interval [a,b], this construction was introduced in 1970 by Alan Day to give a simple solution of the word problem in free lattices. Since then, this construction has found various other applications in lattice theory. In the present paper, it is shown that doubling a convex subset preserves the following generalized semidistributivity condition: (C) for all a,b,c\(\in L\), if \(ab=ac\) and \(c+b=c+a\) then \(a\leq c.\) Furthermore, varieties of lattices in which every lattice satisfies (C) are characterized equationally. These results are analogues of a classic series of theorems about doubling lower pseudo-intervals \(I=\cup [a,b_ i]\), i.e., unions of intervals with a common least element, or upper pseudo-intervals, defined dually.
reviewed by: Günther Eigenthaler
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 4137827
zbMATH Keywords: interval construction
zbMATH Keywords: semidistributive lattice
MaRDI profile type: Publication
cites work: A Simple Solution to the Word Problem for Lattices
cites work: Splitting lattices generate all lattices
cites work: Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices
cites work: Lattice varieties covering the smallest non-modular variety
cites work: Finite Sublattices of a Free Lattice
full work available at URL: https://doi.org/10.1007/bf02034334
OpenAlex ID: W2028101627
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description: scientific article; zbMATH DE number 4137827