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The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states

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Abstract

We study remarkable sub-lattice effect algebras of Archimedean atomic lattice effect algebras E, namely their blocks M, centers C(E), compatibility centers B(E) and sets of all sharp elements S(E) of E. We show that in every such effect algebra E, every atomic block M and the set S(E) are bifull sub-lattice effect algebras of E. Consequently, if E is moreover sharply dominating then every atomic block M is again sharply dominating and the basic decompositions of elements (BDE of x) in E and in M coincide. Thus in the compatibility center B(E) of E, nonzero elements are dominated by central elements and their basic decompositions coincide with those in all atomic blocks and in E. Some further details which may be helpful under answers about the existence and properties of states are shown. Namely, we prove the existence of an (o)-continuous state on every sharply dominating Archimedean atomic lattice effect algebra E with \(B(E)\not =C(E).\) Moreover, for compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved. Further, we prove “State smearing theorem” for these lattice effect algebras.

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Acknowledgments

J. Paseka gratefully acknowledges Financial Support of the Ministry of Education of the Czech Republic under the project MSM0021622409. Z. Riečanová was supported by the Slovak Resaerch and Development Agency under the contract No. APVV–0375–06 and the grant VEGA-2/0032/09 of MŠ SR.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Paseka

  2. Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19, Bratislava, Slovak Republic

    Zdenka Riečanová

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  1. Jan Paseka
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  2. Zdenka Riečanová
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Correspondence to Jan Paseka.

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Paseka, J., Riečanová, Z. The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states. Soft Comput 15, 543–555 (2011). https://doi.org/10.1007/s00500-010-0561-7

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