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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)].

##### MSC:
 06D15 Pseudocomplemented lattices 06A11 Algebraic aspects of posets 06E05 Structure theory of Boolean algebras
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##### References:
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