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The spectrum of a finite pseudocomplemented lattice. (English) Zbl 1209.06004
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 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 G. Grätzer [Proc. Am. Math. Soc. 43, 269–271 (1974; Zbl 0292.06003)].

06D15 Pseudocomplemented lattices
06A11 Algebraic aspects of posets
06E05 Structure theory of Boolean algebras
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[1] Frink O. (1962) Pseudo-complements in semi-lattices. Duke Math. J. 29: 505–514 · Zbl 0114.01602
[2] Glivenko V. (1929) Sur quelques points de la logique de M. Brouwer. Bull. Acad. Sci. Belgique 15: 183–188 · JFM 55.0030.05
[3] Grätzer, G.: Lattice Theory. First Concepts and Distributive Lattices. W. H. Freeman and Co., San Francisco (1971) Softcover edition, Dover Publications (2008) · Zbl 0232.06001
[4] Grätzer G. (1974) A property of transferable lattices. Proc. Amer. Math. Soc. 43: 269–271 · Zbl 0292.06003
[5] Grätzer, G.: General Lattice Theory, 2nd edn. New appendices by the author with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung, and R. Wille. Birkhäuser, Basel (1998) Softcover edition, Birkhuser Verlag, Basel-Boston-Berlin (2003) Reprinted July, 2007.
[6] Grätzer, G.: The Congruences of a Finite Lattice. A Proof-by-Picture Approach. Birkhäuser, Boston (2006) · Zbl 1106.06001
[7] Katriňák T. (1981) A new proof of the Glivenko-Frink theorem. Bull. Soc. Roy. Sci. Liége 50: 171
[8] Katriňák T., Mederly P. (1974) Construction of modular p-algebras. Algebra Universalis 4: 301–315 · Zbl 0316.06005
[9] Riecanová, Z.: Existence of states on effect algebras and an open problem by George Grätzer. Summer School on Algebra and Ordered Sets 2007 (Conference in honor of the 70th birthday of Tibor Katriňák), Hotel Partizán, Tále, Slovakia, 2–7 September 2007. http://www.uvv.umb.sk/algebra2007/
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