The spectrum of a finite pseudocomplemented lattice
type: Item
instance of: scholarly article
title: The spectrum of a finite pseudocomplemented lattice
zbMATH Open document ID: 1209.06004
published in: Algebra Universalis
publication date: 2010-01-04T00:00:00Z
review text: Let \(L\) be a pseudocomplemented lattice, then every interval \([0,a]\) of \(L\) is also pseudocomplemented. So, by Glivenko's theorem, the set \(S(a)\) of all pseudocomplements in \([0,a]\) forms a Boolean lattice. Let \(L\) be a finite pseudocomplemented lattice and suppose that \(S(1)\) has exactly \(n\) atoms. Let \(B_i\) denote the finite Boolean algebra with \(i\) atoms, then the spectrum of \(L\) is the sequence \((s_0,x_1,\dots, s_n)\), where \(s_i=|\{a\in L\mid S(a)\cong B_i\}|\). Clearly, \(s_0+ s_1+\cdots+ s_n=|L|\) and \(s_0= 1\). The main result of the paper is the following theorem: A sequence \((1,s_1,\dots, s_n)\) of positive integers is the spectrum of a finite pseudocomplemented lattice if and only if the inequality \({n\choose i}\leq s_i\) holds for all \(1\leq i\leq n\). This result solves a problem raised in \textit{G. Grätzer}'s book [Lattice theory. First concepts and distributive lattices. San Francisco: Freeman (1971; Zbl 0232.06001)]. The proof uses an induction argument based on the method of ``doubling an element'' which is contained in an earlier paper of \textit{G. Grätzer} [Proc. Am. Math. Soc. 43, 269--271 (1974; Zbl 0292.06003)].
reviewed by: Günther Eigenthaler
Mathematics Subject Classification ID: 06D15
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 06E05
zbMATH DE Number: 5652299
zbMATH Keywords: pseudocomplemented lattice
zbMATH Keywords: Glivenko's theorem
zbMATH Keywords: Boolean algebra
author: G. Grätzer
author: David S. Gunderson
author: Robert W. Quackenbush
MaRDI profile type: Publication
full work available at URL: s00012-009-0027-x
OpenAlex ID: W2057414399
cites work: Pseudo-complements in semi-lattices
cites work: Q5639727
cites work: A Property of Transferable Lattices
cites work: Q5715576
cites work: Construction of modular p-algebras
DOI: 10.1007/S00012-009-0027-X
description: scientific article; zbMATH DE number 5652299