Geodesic
Generalized homogeneous, prelattice and MV-effect algebras
Kybernetika, Tome 41 (2005) no. 2, pp. 129-142 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra $P$ are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra $P$ is a union of generalized MV-effect algebras and every generalized homogeneous effect algebra is a union of its maximal sub-generalized effect algebras with hereditary Riesz decomposition property (blocks). Properties of sharp elements, the center and center of compatibility of $P$ are shown. We prove that on every generalized MV-effect algebra there is a bounded orthogonally additive measure.
We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra $P$ are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra $P$ is a union of generalized MV-effect algebras and every generalized homogeneous effect algebra is a union of its maximal sub-generalized effect algebras with hereditary Riesz decomposition property (blocks). Properties of sharp elements, the center and center of compatibility of $P$ are shown. We prove that on every generalized MV-effect algebra there is a bounded orthogonally additive measure.
Classification : 03B50, 03G12, 03G25, 06D35, 81P10
Keywords: effect algebra; generalized effect algebra; generalized MV- effect algebra; prelattice and homogeneous generalized effect algebra 
@article{KYB_2005_41_2_a2,
     author = {Rie\v{c}anov\'a, Zdenka and Marinov\'a, Ivica},
     title = {Generalized homogeneous, prelattice and {MV-effect} algebras},
     journal = {Kybernetika},
     pages = {129--142},
     year = {2005},
     volume = {41},
     number = {2},
     mrnumber = {2138764},
     zbl = {1249.03122},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2005_41_2_a2/}
}
TY  - JOUR
AU  - Riečanová, Zdenka
AU  - Marinová, Ivica
TI  - Generalized homogeneous, prelattice and MV-effect algebras
JO  - Kybernetika
PY  - 2005
SP  - 129
EP  - 142
VL  - 41
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/KYB_2005_41_2_a2/
LA  - en
ID  - KYB_2005_41_2_a2
ER  - 
%0 Journal Article
%A Riečanová, Zdenka
%A Marinová, Ivica
%T Generalized homogeneous, prelattice and MV-effect algebras
%J Kybernetika
%D 2005
%P 129-142
%V 41
%N 2
%U http://geodesic.mathdoc.fr/item/KYB_2005_41_2_a2/
%G en
%F KYB_2005_41_2_a2
Riečanová, Zdenka; Marinová, Ivica. Generalized homogeneous, prelattice and MV-effect algebras. Kybernetika, Tome 41 (2005) no. 2, pp. 129-142. http://geodesic.mathdoc.fr/item/KYB_2005_41_2_a2/

[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), 1325–1346 | DOI | MR

[4] Gudder S. P.: D-algebras. Found. Phys. 26 (1996), 813–822 | MR

[5] Hájek P.: Mathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht 1998 | MR

[6] Hedlíková J., Pulmannová S.: Generalized difference posets and orthoalgebras. Acta Math. Univ. Comenianae 45 (1996), 247–279 | MR | Zbl

[7] Janowitz M. F.: A note on generalized orthomodular lattices. J. Natur. Sci. Math. 8 (1968), 89–94 | MR | Zbl

[8] Jenča G.: Blocks of homogeneous effect algebras. Bull. Austral. Math. Soc. 64 (2001), 81–98 | DOI | MR | Zbl

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

[10] Kalmbach G.: Orthomodular Lattices. Academic Press, London – New York 1983 | MR | Zbl

[11] Kôpka F.: Compatibility in D-posets. Internat. J. Theor. Phys. 34 (1995), 1525–1531 | DOI | MR | Zbl

[12] Kôpka F., Chovanec F.: Boolean D-posets. Tatra Mt. Math. Publ. 10 (1997), 183–197 | MR | Zbl

[14] Riečanová Z.: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theor. Phys. 38 (1999), 3209–3220 | DOI | MR | Zbl

[15] Riečanová Z.: Compatibilty and central elements in effect algebras. Tatra Mt. Math. Publ. 16 (1999), 151–158

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

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

[18] Wilce A.: Perspectivity and congruence in partial Abelian semigroups. Math. Slovaca 48 (1998), 117–135 | MR | Zbl