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

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Authors

PASEKA Jan RIEČANOVÁ Zdenka

Year of publication 2011
Type Article in Periodical
Magazine / Source Soft computing
MU Faculty or unit

Faculty of Science

Citation
Web http://www.springerlink.com/content/t44107u38q472u54/about/
Doi http://dx.doi.org/10.1007/s00500-010-0561-7
Field General mathematics
Keywords Non-classical logics; MV-algebras; Sharply dominating lattice effect algebras; Basic decomposition of elements; Bifull sub-lattice effect algebras; States
Description 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 equal 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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