Abstract
In this article, we show that certain generalized boolean subalgebras of the exocenter of a generalized effect algebra (GEA) determine hull systems on the GEA in a manner analogous to the determination of a hull mapping on an effect algebra (EA) by its set of invariant elements. We show that a hull system on a GEA E induces a hull mapping on each interval E[0, p] in E, and, using hull systems, we identify certain special elements of E (e.g., η-subcentral elements, η-monads, and η-dyads). We also extend the type-decomposition theory for EAs to GEAs.
Similar content being viewed by others
References
Chevalier G: Around the relative center property in orthomodular lattices. Proc. Amer. Math. Soc. 112, 935–948 (1991)
Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000)
Foulis D.J., Bennett M.K: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)
Foulis D.J., Pulmannová S: Type-decomposition of an effect algebra. Found. Phys. (Mittelstaedt Festschrift) 40, 1543–1565 (2010)
Foulis D.J., Pulmannová S: Centrally orthocomplete effect algebras. Algebra Universalis 64, 283–307 (2010)
Foulis D.J., Pulmannová S: Hull mappings and dimension effect algebras. Math. Slovaca 61, 485–522 (2011)
Foulis D.J., Pulmannová S: The exocenter of a generalized effect algebra. Rep. Math. Phys. 68, 347–371 (2011)
Foulis, D.J., Pulmannová, S.: The center of a generalized effect algebra. To appear in Demonstratio Math. 47 (2014)
Foulis, D.J., Pulmannová, S.: Dimension theory for generalized effect algebras. Algebra Universalis (to appear)
Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Mathematical Surveys and Monographs, vol. 20. American Mathematical Society, Providence (1986)
Grätzer, G.: General Lattice Theory. Academic Press, New York (1978)
Greechie R.J., Foulis D.J., Pulmannová S: The center of an effect algebra. Order 12, 91–106 (1995)
Hedlíková, J., Pulmannová, S.J.: Generalized difference posets and orthoalgebras. Acta Math. Univ. Comenian. (N.S.) 65, 247–279 (1996)
Jenča G: Subcentral ideals in generalized effect algebras. Internat. J. Theoret. Phys. 39, 745–755 (2000)
Jenča G: A Cantor-Bernstein type theorem for effect algebras. Algebra Universalis 48, 399–411 (2002)
Jenča G., Pulmannová S: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Universalis 47, 443–477 (2002)
Kalmbach G., Riečanová Z: An axiomatization for abelian relative inverses. Demonstratio Math. 27, 535–537 (1994)
Loomis, L.H.: The lattice theoretic background of the dimension theory of operator algebras. Mem. Amer. Math. Soc. 18 (1955)
Mayet-Ippolito A: Generalized orthomodular posets. Demonstratio Math. 24, 263–274 (1991)
Polakovič M: Generalized effect algebras of bounded positive operators defined on Hilbert spaces. Rep. Math. Phys. 68, 241–250 (2011)
Polakovič M., Riečanová Z: Generalized effect algebras of positive operators densely defined on Hilbert spaces. Internat. J. Theoret. Phys. 50, 1167–1174 (2011)
Pulmannová S., Vinceková E: Riesz ideals in generalized effect algebras and in their unitizations. Algebra Universalis 57, 393–417 (2007)
Pulmannová S., Vinceková E: Remarks on the order for quantum observables. Math. Slovaca 57, 589–600 (2007)
Riečanová Z: Subalgebras, intervals, and central elements of generalized effect algebras. Internat. J. Theoret. Phys. 38, 3209–3220 (1999)
Riečannová, Z., Zajac, M., Pulmannová, S.: Effect algebras of positive linear operators densely defined on Hilbert spaces. Rep. Math. Phys. 68, 261–270 (2011)
Tkadlec, J.: Atomistic and orthoatomistic effect algebras. J. Math. Phys. 49, 5 pp. (2008)
Wilce A: Perspectivity and congruence in partial abelian semigroups. Math. Slovaca 48, 117–135 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by F. Wehrung.
The second author was supported by ERDF OP R&D metaQUTE ITMS 26240120022 and grant VEGA 2/0059/12, and by the Slovak Research and Development Agency under the contract APVV- 0178-11.
Rights and permissions
About this article
Cite this article
Foulis, D.J., Pulmannová, S. Hull determination and type decomposition for a generalized effect algebra. Algebra Univers. 69, 45–81 (2013). https://doi.org/10.1007/s00012-012-0214-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-012-0214-z