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    Li, Wei Lai, Hongliang and Zhang, Dexue 2016. Yoneda completeness and flat completeness of ordered fuzzy sets. Fuzzy Sets and Systems,


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    Davey, B. A. Haviar, M. and Priestley, H. A. 2007. Boolean Topological Distributive Lattices and Canonical Extensions. Applied Categorical Structures, Vol. 15, Issue. 3, p. 225.


    Erné, Marcel 1996. Convergence structures induced by scales. Topology and its Applications, Vol. 73, Issue. 3, p. 267.


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    Priestley, H.A. 1994. Spectral sets. Journal of Pure and Applied Algebra, Vol. 94, Issue. 1, p. 101.


    Riecanová, Zdenka 1992. Topological and order-topological orthomodular lattices. Bulletin of the Australian Mathematical Society, Vol. 46, Issue. 03, p. 509.


    Erné, Marcel 1981. Continuous Lattices.


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Order-Topological lattices

  • Marcel Erné (a1)
  • DOI: http://dx.doi.org/10.1017/S0017089500003980
  • Published online: 01 May 2009
Abstract

The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].

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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.

6.E. E. Floyd , Boolean algebras with pathological order topologies, Pacific J. Math. 5 (1955), 687689.

12.H.-J. Kowalsky , Beiträge zur topologischen Algebra, Math. Nachr. 11 (1954), 143185.

14.J. Schmidt , Beiträge zur Filtertheorie I, Math. Nachr. 7 (1952), 359378.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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