Finite pseudocomplemented lattices: The spectra and the Glivenko congruence
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.
scientific article; zbMATH DE number 5977541