Geodesic
Archimedean atomic lattice effect algebras in which all sharp elements are central
Kybernetika, Tome 42 (2006) no. 2, pp. 143-150 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
Classification : 03G10, 03G12, 03G25, 06D35, 81P10
Keywords: lattice effect algebra; sharp and central element; block; state; subdirect decomposition; MacNeille completion 
@article{KYB_2006_42_2_a1,
     author = {Rie\v{c}anov\'a, Zdenka},
     title = {Archimedean atomic lattice effect algebras in which all sharp elements are central},
     journal = {Kybernetika},
     pages = {143--150},
     year = {2006},
     volume = {42},
     number = {2},
     mrnumber = {2241781},
     zbl = {1249.03121},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2006_42_2_a1/}
}
TY  - JOUR
AU  - Riečanová, Zdenka
TI  - Archimedean atomic lattice effect algebras in which all sharp elements are central
JO  - Kybernetika
PY  - 2006
SP  - 143
EP  - 150
VL  - 42
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/KYB_2006_42_2_a1/
LA  - en
ID  - KYB_2006_42_2_a1
ER  - 
%0 Journal Article
%A Riečanová, Zdenka
%T Archimedean atomic lattice effect algebras in which all sharp elements are central
%J Kybernetika
%D 2006
%P 143-150
%V 42
%N 2
%U http://geodesic.mathdoc.fr/item/KYB_2006_42_2_a1/
%G en
%F KYB_2006_42_2_a1
Riečanová, Zdenka. Archimedean atomic lattice effect algebras in which all sharp elements are central. Kybernetika, Tome 42 (2006) no. 2, pp. 143-150. http://geodesic.mathdoc.fr/item/KYB_2006_42_2_a1/

[1] Chang C. C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467–490 | DOI | MR | Zbl

[2] Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht – Boston – London and Ister Science, Bratislava 2000 | MR

[3] Foulis D., Bennett M. K.: Effect algebras and unsharp quantum logics. Found. Phys. 24 (1994), 1331–1352 | DOI | MR

[4] Greechie R. J.: Orthomodular lattices admitting no states. J. Combin. Theory Ser. A 10 (1971), 119–132 | DOI | MR | Zbl

[5] Greechie R. J., Foulis, D., Pulmannová S.: The center of an effect algebra. Order 12 (1995), 91–106 | DOI | MR | Zbl

[6] Jenča G., Riečanová Z.: On sharp elements in lattice effect algebras. BUSEFAL 80 (1999), 24–29

[7] Kôpka F., Chovanec F.: $D$-posets. Math. Slovaca 44 (1994), 21–34 | MR | Zbl

[8] Riečanová Z.: MacNeille completions of $D$-posets and effect algebras. Internat. J. Theor. Phys. 39 (2000), 859–869 | DOI | MR | Zbl

[9] Riečanová Z.: Generalization of blocks for $D$-lattices and lattice ordered effect algebras. Internat. J. Theor. Phys. 39 (2000), 231–237 | DOI | MR | Zbl

[10] Riečanová Z.: Archimedean and block-finite lattice effect algebras. Demonstratio Math. 33 (2000), 443–452 | MR

[11] Riečanová Z.: Orthogonal sets in effect algebras. Demonstratio Math. 34 (2001), 3, 525–532 | MR

[12] Riečanová Z.: Proper effect algebras admitting no states. Internat. J. Theor. Phys. 40 40 (2001), 1683–1691 | DOI | MR | Zbl

[13] Riečanová Z.: Smearings of states defined on sharp elements onto effect algebras. Internat. J. Theor. Phys. 41 (2002), 1511–1524 | DOI | MR

[14] Riečanová Z.: Continuous effect algebra admitting order-continuous states. Fuzzy Sets and Systems 136 (2003), 41–54 | DOI | MR

[15] Riečanová Z.: Distributive atomic effect algebras. Demonstratio Math. 36 (2003), 247–259 | MR

[16] Riečanová Z.: Subdirect decompositions of lattice effect algebras. Internat. J. Theor. Phys. 42 (2003), 1425–1433 | DOI | MR

[17] Riečanová Z.: Modular atomic effect algebras and the existence of subadditive states. Kybernetika 40 (2004), 459–468 | MR

[18] Schmidt J.: Zur Kennzeichnung der Dedekind–MacNeilleschen Hulle einer geordneten Menge. Arch. Math. 17 (1956), 241–249 | DOI | MR