Finite pseudocomplemented lattices: The spectra and the Glivenko congruence
type: Item
instance of: scholarly article
title: Finite pseudocomplemented lattices: The spectra and the Glivenko congruence
zbMATH Open document ID: 1231.06015
published in: Algebra Universalis
publication date: 2011-11-23T00:00:00Z
review text: Given a pseudocomplemented semilattice \(P\) and \(a\in P\), the principal ideal \((a]\) is known to be a pseudocomplemented semilattice. The corresponding Boolean lattices of closed elements will be denoted by \(S(P)\) and \(S(a)\), respectively. Let \(L\) be a finite pseudocomplemeted lattice \(L\), where \(S(L)\) has exactly \(n\) atoms, and let \(B_i\) be the finite Boolean lattice with \(i\) atoms. The Boolean spectrum of \(L\) is the sequence \({\mathbf s}=(s_0,\dots,s_n)\), where \[ s_i=|\{a\in L: S(a)\cong B_i\}|. \] Recently, \textit{G. Grätzer, D. S. Gunderson} and \textit{R. W. Quackenbush} have characterized the spectra of finite pseudocomplemented semilattices [``The spectrum of a finite pseudocomplemented lattice'', Algebra Univers. 61, No. 3--4, 407--411 (2009; Zbl 1209.06004)]. In the paper under review a tight connection between the spectra and the Glivenko congruence of finite pseudocomplemented semilattices is discussed. Also the spectra of finite Stone lattices are characterized.
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06D15
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 06E05
zbMATH DE Number: 5977541
zbMATH Keywords: pseudocomplemented lattice
zbMATH Keywords: pseudocomplemented semilattice
zbMATH Keywords: Glivenko congruence
zbMATH Keywords: Boolean lattice
author: Tibor Katriňák
author: J. Guričan
MaRDI profile type: Publication
full work available at URL: s00012-011-0145-0
OpenAlex ID: W2024023804
cites work: Q5532570
cites work: Pseudo-complements in semi-lattices
cites work: Q5639727
cites work: Q4399900
cites work: The spectrum of a finite pseudocomplemented lattice
cites work: Q5328221
cites work: Subdirectly irreducible modular p-algebras
cites work: Construction of modular p-algebras
DOI: 10.1007/S00012-011-0145-0
description: scientific article; zbMATH DE number 5977541