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Intervals in generalized effect algebras

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Abstract

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Riečanová and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.

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Acknowledgments

Our most sincere thanks go to Silvia Pulmannová for valuable comments. Jiří Janda kindly acknowledges the support by Masaryk University, grant 0964/2009 and ESF Project CZ.1.07 /2.3.00/20.0051 Algebraic Methods in Quantum Logic of the Masaryk University. Zdenka Riečanová kindly acknowledges the support by the Science and Technology Assistance Agency under the contract APVV-0178-11 Bratislava SR, and VEGA-grant of MŠ SR No. 1/0297/11.

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Authors and Affiliations

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

    Jiří Janda

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

    Zdenka Riečanová

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  1. Jiří Janda
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  2. Zdenka Riečanová
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Correspondence to Jiří Janda.

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Communicated by L. Spada.

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Janda, J., Riečanová, Z. Intervals in generalized effect algebras. Soft Comput 18, 413–418 (2014). https://doi.org/10.1007/s00500-013-1083-x

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  • DOI: https://doi.org/10.1007/s00500-013-1083-x

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