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

J 2011

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

PASEKA, Jan and Zdenka RIEČANOVÁ

Basic information

Original name

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

Authors

PASEKA, Jan (203 Czech Republic, guarantor, belonging to the institution) and Zdenka RIEČANOVÁ (703 Slovakia)

Edition

Soft computing, Springer-Verlag GmbH, 2011, 1432-7643

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 1.880

RIV identification code

RIV/00216224:14310/11:00055100

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1007/s00500-010-0561-7

UT WoS

000287451000012

Keywords in English

Non-classical logics; MV-algebras; Sharply dominating lattice effect algebras; Basic decomposition of elements; Bifull sub-lattice effect algebras; States

Abstract

V originále

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.

Links

MSM0021622409, plan (intention)

Name: Matematické struktury a jejich fyzikální aplikace

Investor: Ministry of Education, Youth and Sports of the CR, Mathematical structures and their physical applications

Citovat

PASEKA, Jan and Zdenka RIEČANOVÁ. The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states. Soft computing. Springer-Verlag GmbH, 2011, vol. 15, No 3, p. 543-555. ISSN 1432-7643. Available from: https://dx.doi.org/10.1007/s00500-010-0561-7.

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   author = {Paseka, Jan and Riečanová, Zdenka},
   article_number = {3},
   doi = {http://dx.doi.org/10.1007/s00500-010-0561-7},
   keywords = {Non-classical logics; MV-algebras; Sharply dominating lattice effect algebras; Basic decomposition of elements; Bifull sub-lattice effect algebras; States},
   language = {eng},
   issn = {1432-7643},
   journal = {Soft computing},
   title = {The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states},
   url = {http://www.springerlink.com/content/t44107u38q472u54/about/},
   volume = {15},
   year = {2011}
}

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JF  - Soft computing
VL  - 15
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PB  - Springer-Verlag GmbH
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KW  - Non-classical logics
KW  - MV-algebras
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KW  - Basic decomposition of elements
KW  - Bifull sub-lattice effect algebras
KW  - States
UR  - http://www.springerlink.com/content/t44107u38q472u54/about/
L2  - http://www.springerlink.com/content/t44107u38q472u54/about/
N2  - 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.
ER  -

PASEKA, Jan and Zdenka RIEČANOVÁ. The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states. \textit{Soft computing}. Springer-Verlag GmbH, 2011, vol.~15, No~3, p.~543-555. ISSN~1432-7643. Available from: https://dx.doi.org/10.1007/s00500-010-0561-7.

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