Abstract
We study congruences on pseudoeffect algebras, which were recently introduced as a non-commutative generalization of effect algebras. We introduce ideals for these algebras and give a sufficient condition for an ideal to determine a congruence. Furthermore, states on pseudoeffect algebras are considered. It is shown that any interval pseudoeffect algebra maps homomorphically into an effect algebra whose states are in a one-to-one correspondence to the states of the original algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
D. J. Foulis and M. K. Bennett, “Effect algebras and unsharp quantum logics,” Found. Phys. 24 (1994) 1325–1346.
P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, Berlin, 1995).
F. Kôpka and F. Chovanec, “D-posets,” Math. Slovaca 44 (1994) 21–34.
R. Giuntini and H. Greuling, “Towards a formal language for unsharp properties,” Found. Phys. 19 (1989) 931–945.
K. Ravindran, “On a Structure Theory of Effect Algebras,” Ph.D. Thesis (Kansas State University, Manhattan, 1996).
R. Cignoli, I. M. L. D'Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning (Kluwer Academic, Dordrecht, 2000).
A. Dvurečenskij and T. Vetterlein, “Pseudoeffect algebras. I. Basic properties,” Inter. J. Theor. Phys. 40 (2001) 685–701.
G. Georgescu and A. Iorgulescu, “Pseudo-MV algebras,” Mult. Val. Logic, 6 (2001) 95–135.
A. Dvurečenskij and T. Vetterlein, “Pseudoeffect algebras. II. Group representations,” Inter. J. Theor. Phys. 40 (2001) 703–726.
R. Baudot, “Non-commutative logic programming: The language NoClog,” in Symposium on Logic in Computer Science (Santa Barbara, 2000), pp. 3–9.
S. Gudder and G. Nagy, “Sequentially independent effects,” submitted.
D. J. Foulis, “Sequential probability models and transition probabilities,” Atti Semin. Mat. Fis. Univ. Modena, to appear.
S. Gudder and S. Pulmannová, “Quotients of partial Abelian monoids,” Algebra Universalis 38 (1997) 395–421.
M. K. Bennett and D. J. Foulis, “Interval and scale effect algebras,” Adv. Appl. Math. 91 (1997) 200–215.
G. Grätzer, Universal Algebra (Springer, New York, 1979).
L. Fuchs, Partially Ordered Algebraic Systems (Pergamon, Oxford, 1963).
M. R. Darnel, Theory of Lattice-Ordered Groups (Marcel Dekker, New York, 1995).
A. M. W. Glass, Partially Ordered Groups (Series in Algebra 7) (World Scientific, Singapore, 1999).
V. M. Kopytov and N. Ya. Medvedev, The Theory of Lattice-Ordered Groups (Kluwer Academic, Dordrecht, 1994).
K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation (Mathematical Surveys and Monographs 20) (American Mathematical Society, Providence, 1986).
A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures (Kluwer Academic, Dordrecht; Ister Science, Bratislava, 2000).
O. Wyler, “Clans,” Compos. Math. 17 (1966) 172–189.
H. Schubert, Topologie (Teubner, Stuttgart, 1975).
M. Navara, “An orthomodular lattice admitting no group-valued measure,” Proc. Amer. Math. Soc. 122 (1994) 7–12.
H. Weber, “There are orthomodular lattices without non-trivial group-valued states: A computer-based construction,” J. Math. Anal. Appl. 183 (1994) 89–93.
A. Dvurečenskij, “Pseudo-MV algebras are intervals in ℓ-groups,” J. Austr. Math. Soc., Ser. A, to appear.
A. Dvurečenskij, “States on pseudo-MV algebras,” Studia Logica, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dvurecenskij, A., Vetterlein, T. Congruences and States on Pseudoeffect Algebras. Found Phys Lett 14, 425–446 (2001). https://doi.org/10.1023/A:1015561420306
Published:
Issue Date:
DOI: https://doi.org/10.1023/A:1015561420306