Abstract
The notion of a Sasaki projectionon an orthomodular lattice is generalized to a mapping Φ: E × E → E, where E is an effect algebra. If E is lattice ordered and Φ is symmetric, then E is called a Φ-symmetric effect algebra.This paper launches a study of such effect algebras. In particular, it is shown that every interval effect algebra with a lattice-ordered ambient group is Φ-symmetric, and its group is the one constructed by Ravindran in his proof that every effect algebra that has the Riesz decomposition property is an interval algebra. It is shown that the doubling construction introduced in the paper is connected to the conditional event algebrasof Goodman, Nguyen, and Walker.
Similar content being viewed by others
References
M. K. Bennett and D. J. Foulis, “Interval algebras and unsharp quantum logics,” preprint. University of Massachusetts, 1994.
T. S. Blyth and M. F. Janowitz,Residuation Theory (Pergamon, New York, 1972).
G. Cattaneo and G. Nistico, “Brouwer-Zadeh posets and three-valued Lukasiewicz posets,”Int. J. Fuzzy Sets Syst. 33, 165–190 (1989).
J. C. Derderian, “Residuated mappings,”Pacific J. Math. 20, 35–44 (1976).
D. J. Foulis, “A note on orthomodular lattices,”Portugaliae Mathematica 21, 65–72 (1962).
D. J. Foulis and M. K. Bennett, “Effect algebras and unsharp quantum logics,”Found. Phys. 24 (10), 1325–1346 (1994).
D. J. Foulis, R. J. Greechie, and M. K. Bennett, “Sums and products of interval algebras,”Int. J. Theor. Phys. 33(11), 2119–2136 (1994).
K. R. Goodearl,Partially Ordered Abelian Groups with Interpolation (American Mathematical Society Surveys and Monographs, No. 20) (American Mathematical Society, Providence, Rhode Island, 1986).
I. R. Goodman, “Toward a comprehensive theory of linguistic and probabilistic evidence: Two new approaches to conditional event algebra,”IEEE Trans. Systems, Man. Cybern. 24(12), 1685–1698 (1994).
R. J. Greechie and D. J. Foulis, “The transition to effect algebras,”Int. J. Theor. Phys. 34(8), 1–14 (1995).
R. J. Greechie, D. J. Foulis, and S. Pulmannová, “The center of an effect algebra,”Order 12, 91–106 (1995).
S. P. Gudder, “Effect algebras and tensor products of S-sets,”Int. J. Theor. Phys., to appear, (1995).
S. P. Gudder, “Chain tensor products and interval effect algebras,” preprint, University of Denver, 1995.
G. Kalmbach,Orthomodular Lattices (Academic, New York, 1983).
K. Ravindran, “On some structural aspects of effect algebras,” preprint; Louisiana Tech University, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bennett, M.K., Foulis, D.J. Phi-symmetric effect algebras. Found Phys 25, 1699–1722 (1995). https://doi.org/10.1007/BF02057883
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02057883