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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
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