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Some Finiteness Conditions for Orthomodular Lattices

Published online by Cambridge University Press:  20 November 2018

Günter Bruns  and
Richard Greechie
Günter Bruns
Affiliation:
McMaster University, Hamilton, Ontario
Richard Greechie
Affiliation:
Kansas State University, Manhattan, Kansas
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Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:

A n : L has at most n blocks,

B n : there exists a covering of L by at most n blocks,

C n: the set ﹛C(x)| x ∈ L﹜ has at most n elements,

D n : out of any n + 1 elements of L at least two commute.

We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ n An , B ⇔ ∃ n Bn , C ⇔ ∃ n Cn and D ⇔ ∃ n Dn .

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 535 - 549
Copyright
Copyright © Canadian Mathematical Society 1982

References

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