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    Harding, John 2012. A Boolean topological orthomodular poset. Algebra universalis, Vol. 68, Issue. 3-4, p. 193.


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  • Bulletin of the Australian Mathematical Society, Volume 46, Issue 3
  • December 1992, pp. 509-518

Topological and order-topological orthomodular lattices

  • Zdenka Riecanová (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700012168
  • Published online: 01 April 2009
Abstract

The necessary and sufficient conditions for atomic orthomodular lattices to have the MacNeille completion modular, or (o)-continuous or order topological, orthomodular lattices are proved. Moreover we show that if in an orthomodular lattice the (o)-convergence of filters is topological then the (o)-convergence of nets need not be topological. Finally we show that even in the case when the MacNeille completion of an orthomodular lattice L is order-topological, then in general the (o)-convergence of nets in does not imply their (o)-convergence in L. (This disproves, also for the orthomodular and order-topological case, one statement in G.Birkhoff's book.)

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]G. Bruns , R.J. Greechie , J. Harding , and M. Roddy , ‘Completions of orthomodular lattices’, Order 7 (1990), 6776.

[5]N. Funayama , ‘On the completion by cuts of a distributive lattice’, Proc. Imp. Acad. Tokyo 20 (1944).

[6]M.F. Janowitz , ‘Indexed orthomodular lattices’, Math. Z. 119 (1971), 2832.

[8]J. Kaplansky , ‘Any orthocomplemented complete modular lattice is continuous geometry’, Ann. of Math. 61 (1955), 524541.

[11]M. Erné and S. Weck , ‘Order convergence in lattices’, Rocky Mountain J. Math. 10 (1980), 805818.

[15]J. Schmidt , ‘Zur Kennzeichnung der Dedekind MacNeilleschen Hulle einer Geordneten Menge’, Arch. Math. 7 (1956), 241249.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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