| Title:
|
Generalized homogeneous, prelattice and MV-effect algebras (English) |
| Author:
|
Riečanová, Zdenka |
| Author:
|
Marinová, Ivica |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
41 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
[129]-142 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
effect algebra |
| Keyword:
|
generalized effect algebra |
| Keyword:
|
generalized MV- effect algebra |
| Keyword:
|
prelattice and homogeneous generalized effect algebra |
| MSC:
|
03B50 |
| MSC:
|
03G12 |
| MSC:
|
03G25 |
| MSC:
|
06D35 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1249.03122 |
| idMR:
|
MR2138764 |
| . |
| Date available:
|
2009-09-24T20:07:41Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135646 |
| . |
| Reference:
|
[1] Chang C. C.: Algebraic analysis of many-valued logics.Trans. Amer. Math. Soc. 88 (1958), 467–490 Zbl 0084.00704, MR 0094302, 10.1090/S0002-9947-1958-0094302-9 |
| Reference:
|
[2] Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures.Kluwer Academic Publishers, Dordrecht – Boston – London and Ister Science, Bratislava 2000 MR 1861369 |
| Reference:
|
[3] Foulis D., Bennett M. K.: Effect algebras and unsharp quantum logics.Found. Phys. 24 (1994), 1325–1346 MR 1304942, 10.1007/BF02283036 |
| Reference:
|
[4] Gudder S. P.: D-algebras.Found. Phys. 26 (1996), 813–822 MR 1407184 |
| Reference:
|
[5] Hájek P.: Mathematics of Fuzzy Logic.Kluwer Academic Publishers, Dordrecht 1998 MR 1900263 |
| Reference:
|
[6] Hedlíková J., Pulmannová S.: Generalized difference posets and orthoalgebras.Acta Math. Univ. Comenianae 45 (1996), 247–279 Zbl 0922.06002, MR 1451174 |
| Reference:
|
[7] Janowitz M. F.: A note on generalized orthomodular lattices.J. Natur. Sci. Math. 8 (1968), 89–94 Zbl 0169.02104, MR 0231762 |
| Reference:
|
[8] Jenča G.: Blocks of homogeneous effect algebras.Bull. Austral. Math. Soc. 64 (2001), 81–98 Zbl 0985.03063, MR 1848081, 10.1017/S0004972700019705 |
| Reference:
|
[9] Jenča G., Riečanová Z.: On sharp elements in lattice ordered effect algebras.BUSEFAL 80 (1999), 24–29 |
| Reference:
|
[10] Kalmbach G.: Orthomodular Lattices.Academic Press, London – New York 1983 Zbl 0554.06009, MR 0716496 |
| Reference:
|
[11] Kôpka F.: Compatibility in D-posets.Internat. J. Theor. Phys. 34 (1995), 1525–1531 Zbl 0851.03020, MR 1353696, 10.1007/BF00676263 |
| Reference:
|
[12] Kôpka F., Chovanec F.: Boolean D-posets.Tatra Mt. Math. Publ. 10 (1997), 183–197 Zbl 0915.03052, MR 1469294 |
| Reference:
|
[14] Riečanová Z.: Subalgebras, intervals and central elements of generalized effect algebras.Internat. J. Theor. Phys. 38 (1999), 3209–3220 Zbl 0963.03087, MR 1764459, 10.1023/A:1026682215765 |
| Reference:
|
[15] Riečanová Z.: Compatibilty and central elements in effect algebras.Tatra Mt. Math. Publ. 16 (1999), 151–158 |
| Reference:
|
[16] Riečanová Z.: Generalization of blocks for $D$-lattices and lattice ordered effect algebras.Internat. J. Theor. Phys. 39 (2000), 231–237 Zbl 0968.81003, MR 1762594, 10.1023/A:1003619806024 |
| Reference:
|
[17] Riečanová Z.: Orthogonal sets in effect algebras.Demonstratio Mathematica 34 (2001), 525–532 MR 1853730 |
| Reference:
|
[18] Wilce A.: Perspectivity and congruence in partial Abelian semigroups.Math. Slovaca 48 (1998), 117–135 Zbl 0938.03094, MR 1647650 |
| . |
