Isomorphism theorems on generalized effect algebras based on atoms

Authors:
Jan Paseka;Zdenka Riečanová
Affiliations:
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Janáčkovo nám. 2a, 602 00 Brno, Czech Republic;Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, SK-812 19 Bratislava, Slovak Republic
Venue:
Information Sciences: an International Journal
Year:
2009

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Abstract

A well-known fact is that every generalized effect algebra can be uniquely extended to an effect algebra in which it becomes a sub-generalized effect algebra and simultaneously a proper order ideal, the set-theoretic complement of which is its dual poset. We show that two non-isomorphic generalized effect algebras (even finite ones) may have isomorphic effect algebraic extensions. For Archimedean atomic lattice effect algebras we prove ''Isomorphism theorem based on atoms''. As an application we obtain necessary and sufficient conditions for isomorphism of two prelattice Archimedean atomic generalized effect algebras with common (or isomorphic) effect algebraic extensions.