Abstract
Common generalizations of orthomodular lattices and MV-algebras are lattice effect algebras which may include noncompatible pairs of elements as well as unsharp elements. Thus elements of these structures may be carriers of states, or probability measures, when they represent properties, questions or events with fuzziness, uncertainty or unsharpness. Unbounded versions of these structures (more precisely without top elements) are generalized effect algebras which can be extended onto effect algebras. We touch only a few aspects of these structures. Namely, necessary and sufficient conditions for generalized effect algebras to obtain their effect algebraic extensions lattice ordered or MV-effect algebras. We also give one possible construction of pastings of MV-effect algebras together along an MV-effect algebra to obtain lattice effect algebras. In conclusions we give some applications of presented results about sets of sharp elements, direct and subdirect decompositions of lattice effect algebras and about smearings (resp. the existence) of states an probabilities on them.
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References
Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)
Chovanec, F., Kôpka, F.: Boolean D-posets. Tatra Mt. Math. Publ. 10, 183–197 (1997)
Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht, Ister Science, Bratislava (2000)
Foulis, D., Bennett, M.K.: Effect algebras and unsharp quantum logics. Foundations of Physics 24, 1331–1352 (1994)
Gudder, S.: S-dominating effect algebras. Internat. J. Theor. Phys. 37, 915–923 (1998)
Jenča, G., Riečanová, Z.: On sharp elements in lattice effect algebras. Busefal 80, 24–29 (1999)
Kôpka, F.: Compatibility in D-posets. Internat. J. Theor. Phys. 34, 1525–1531 (1995)
Greechie, R.J.: Orthomodular lattices admitting no states. J. Combin. Theory Ser. A 10, 119–132 (1971)
Kôpka, F., Chovanec, F.: D-posets. Mathematica Slovaca 44, 21–34 (1994)
Navara, M.P., Rogalewicz, V.: The pasting construction for orthomodular posets. Mathematische Nachrichten 154, 157–168 (1991)
Riečanová, Z.: Subalgebras, intervals and central elements of generalized effect algebras. Internat. J. Theor. Phys. 38, 3209–3220 (1999)
Riečanová, Z.: Compatibilty and central elements in effect algebras. Tatra Mt. Math. Publ. 16, 151–158 (1999)
Riečanová, Z.: MacNeille completions of D-posets and effect algebras. Internat. J. Theor. Phys. 39, 859–869 (2000)
Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras. Internat. J. Theor. Phys. 39, 231–237 (2000)
Riečanová, Z.: Archimedean and block-finite lattice effect algebras. Demonstratio Mathematica 33, 443–452 (2000)
Riečanová, Z.: Orthogonal sets in effect algebras. Demonstratio Mathematica 34, 525–532 (2001)
Riečanová, Z.: Proper effect algebras admitting no states. Internat. J. Theor. Phys. 40, 1683–1691 (2001)
Riečanová, Z.: Smearings of states defined on sharp elements onto effect algebras. Internat. J. Theor. Phys. 41, 1511–1524 (2002)
Riečanová, Z.: Continuous effect algebra admitting order-continuous states. Fuzzy Sets and Systems 136, 41–54 (2003)
Riečanová, Z.: Distributive atomic effect algebras. Demonstratio Mathematica 36, 247–259 (2003)
Riečanová, Z.: Subdirect decompositions of lattice effect algebras. Internat. J. Theor. Phys. 42, 1425–1433 (2003)
Riečanová, Z.: Modular atomic effect algebras and the existence of subadditive states. Kybernetika 40, 459–468 (2004)
Riečanová, Z.: Basic decomposition of elements and Jauch-Piron effect algebras. Fuzzy Sets and Systems 155, 138–149 (2005)
Riečanová, Z.: Embeddings of generalized effect algebras into complete effect algebras. Soft Computing 10, 476–482 (2006)
Riečanová, Z.: Pastings of MV-effect algebras. Internat. J. Theor. Phys. 43, 1875–1883 (2004)
Riečanová, Z., Bršel, D.: Contraexamples in difference posets and orthoalgebras. Internat. J. Theor. Phys. 33, 133–141 (1994)
Riečanová, Z., Marinová, I.: Generalized homogeneous, prelattice and MV-effect algebras. Kybernetika 41, 129–141 (2005)
Riečanová, Z., Marinová, I., Zajac, M.: From Real Numbers to Effect Algebras and Families of Fuzzy Sets. In: Begabtenförderung im MINT-Bereich (Mathematik, Informatik, Naturwissenschaften, Technik) ch. 7, band 9, pp. 103–115 (2004)
Schmidt, J.: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hulle einer Geordneten Menge. Archiv d. Math. 7, 241–249 (1956)
Yutani, H.: The class of commutative BCK-algebras is equationally definable. Math. Seminar Notes 5, 207–210 (1977)
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Riečanová, Z., Marinová, I., Zajac, M. (2006). Some Aspects of Lattice and Generalized Prelattice Effect Algebras. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments II. Lecture Notes in Computer Science(), vol 4342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11964810_14
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DOI: https://doi.org/10.1007/11964810_14
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