×

zbMATH — the first resource for mathematics

Finite pseudocomplemented lattices: The spectra and the Glivenko congruence. (English) Zbl 1231.06015
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, G. Grätzer, D. S. Gunderson and 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.

MSC:
06D15 Pseudocomplemented lattices
06A12 Semilattices
06E05 Structure theory of Boolean algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Birkhoff G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967) · Zbl 0153.02501
[2] Frink O.: Pseudo-complements in semi-lattices. Duke Math. J. 29, 505–514 (1962) · Zbl 0114.01602 · doi:10.1215/S0012-7094-62-02951-4
[3] Glivenko V.: Sur quelques points de la logique de M. Brouwer. Bull Acad. Sci. Belgique 15, 183–188 (1929) · JFM 55.0030.05
[4] Grätzer G.: Lattice Theory First. Concepts and Distributive Lattices. W. H. Freeman and Co., San Francisco (1971) · Zbl 0232.06001
[5] Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998) · Zbl 0909.06002
[6] Grätzer G., Gunderson D.S., Quackenbush R.W.: The spectrum of a finite pseudocomplemented lattice. Algebra Universalis 61, 407–411 (2009) · Zbl 1209.06004 · doi:10.1007/s00012-009-0027-x
[7] Jacobson N.: Lectures in Abstract Algebra, vol. I. D. van Nostrand, Princeton (1964) · Zbl 0124.27002
[8] Katriňák T.: Subdirectly irreducible modular p-algebras. Algebra Universalis 2, 166–173 (1972) · Zbl 0258.06005 · doi:10.1007/BF02945024
[9] Katriňák T.: A new proof of the construction theorem for Stone algebras. Proc. Amer. Math. Soc. 40, 75–78 (1973) · Zbl 0266.06005 · doi:10.2307/2038636
[10] Katriňák T., Mederly P.: Construction of modular p-algebras. Algebra Universalis 4, 301–315 (1974) · Zbl 0316.06005 · doi:10.1007/BF02485742
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.