On duality between order and algebraic structures in Boolean systems

We present an extension of the known one-to-one correspondence between Boolean algebras and Boolean rings with unit being two types of Boolean systems endowed with order and algebraic structures, respectively. Two equivalent generalizations of Boolean algebras are discussed. We show that there is a one-to-one correspondence between any of the two mentioned generalized Boolean algebras and Boolean rings without unit.

Full text (pdf)

1. B. Aniszczyk, J. Burzyk, A. Kaminski, Borel and monotone hierarchies and extensions of Renyi probability spaces, Colloq. Math. 51 (1987), 9-25. 
2. L. Beran, Orthomodular Lattices, Algebraic Approach, Academia/Reidel, Praque/Dordrecht 1984.
3. G. Birkho , Lattice Theory, Amer. Math. Soc., Providence, Rhode Island, First edition 1940; Second revised edition 1961; Third edition 1967; Corrected reprint of the third edition 1979.
4. I. Chajda, Pseudosemirings induced by ortholattices, Czechoslovak Math. J. 46
   (1996), 405-411.
5. [5] I. Chajda, G. Eigenthaler, A note on orthopseudorings and Boolean quasirings, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 207 (1998), 83-94.
6. I. Chajda, H. Langer, Ring-like operations in pseudocomplemented semilattices, Discuss. Math. Gen. Algebra Appl. 20 (2000), 87-95.
7. I. Chajda, H. Langer, M. Maczynski, Ring like structures corresponding to generalized orthomodular lattices, Math. Slovaca 54 (2004), 143-150. [8] F. Chovanec, F. Kopka, D-posets, In:
8. K. Engesser, D. M. Gabbay, D. Lehmann (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Structures, 367-428, Elsevier, Amstredam, 2007.
9. A. Dadej, K. Halik, Properties of di erences in B-rings, J. Math. Appl. 34 (2011), 5-13.
10. D. Dorninger, H. Langer, M. Maczynski, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232.
11. D. Dorninger, H. Langer, M. Maczynski, On ring-like structures occurring in axiomatic  quantum mechanics, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279-289.
12. D. Dorninger, H. Langer, M. Maczynski, On ring-like structures induced by Mackey's  probability function, Rep. Math. Phys. 43 (1999), 499-515.
13. D. Dorninger, H. Langer, M. Maczynski, Lattice properties of ring-like quantum logics, Internat. J. Theoret. Phys. 39 (2000), 1015-1026.
14. D. Dorninger, H. Langer, M. Maczynski, Concepts of measures on ring-like quantum logics, Rep. Math. Phys. 47 (2001), 167-176.
15. D. Dorninger, H. Langer, M. Maczynski, Ring-like structures with unique symmetric di erence related to quantum logic, Discuss. Math. Gen. Algebra Appl. 21 (2001), 239-253.
16. R. Fric, Remarks on statistical maps and fuzzy (operational) random variables, Tatra Mt. Math. Publ. 30 (2005), 21-34.
17. R. Fric, Statistical maps: a categorical approach, Math. Slovaca 57 (2007), 41-57.
18. R. Fric, M. Papco, On probability domains, Int. J. Theor. Phys. 49 (2010), 3092-3100.
19. R. Fric, M. Papco, On probability domains II, Int. J. Theor. Phys. 50 (2011), 3778-3786.
20. R. Fric, M. Papco, A categorical approach to probability, Stud. Log. 94 (2010),
    215-230.
21. G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel-Boston-Berlin, First edition 1978; Second edition 1998; Reprint of the second edition 2003.
22. K. Halik, A. Pydo, A note on Sierpinski-Pettis theorems in terms of generalized Boolean -algebras, Scienti c Bulletin of Che lm 2 (2006), 67-72.
23. M. F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci. Math. 8 (1968), 89-94.
24. G. Kalmbach, Orthomodular Lattices, Academic Press, London 1983.
25. A. Kaminski, Distributors and Renyi's theory of conditional probabilities, in: Generalized Functions and Operational Calculus, Proc. Conf., Varna 1975, Publishing House of the Bulg. Acad. of Sciences, So a 1975, 111-121.
26. A. Kaminski, On the Renyi theory of conditional probabilities, Studia Math. 79 (1984),  151-191.
27. A. Kaminski, On extensions of Renyi conditional probability spaces, Colloq. Math. 49 (1985), 267-294.
28. A. Kaminski, Generated -rings and -algebras, Institute of Mathematics Polish Academy of Sciences, Preprint 385, Waraszawa 1987, 1-36.
29. A. Kaminski, Convolution of distributions and Renyi's theory of conditional probabilities, Integral Transforms and Spec. Funct. 6 (1998), 215-222.
30. F. Kopka, D-posets of fuzzy sets, Tatra Mt. Math. Publ. 1 (1992), 83-87.
31. F. Kopka, Quasi product on Boolean D-posets, Int. J. Theor. Phys. 47 (2008), 1625-1632.
32. F. Kopka, F. Chovanec, D-posets, Math. Slovaca 15 (1994), 21-34.
33. M. Papco, On measurable spaces and measurable maps, Tatra Mt. Math. Publ.28 (2004), 125-140.
34. M. Papco, On fuzzy random variables: examples and generalizations, Tatra Mt. Math. Publ. 30 (2005), 175-185.
35. M. Papco, On e ect algebras, Soft Comput. 12 (2008), 373-379.
36. B. J. Pettis, On some theorems of Sierpinski on subalgebras of Boolean -rings,  Bull. Acad. Polon. Sci. 19 (1971), 563-568.
37. A. Renyi, On a new axiomatic theory of probability, Acta Math. Acad. Sci. Hungar. 6 (1955), 285-335.
38. A. Renyi, Probability Theory, Akademiai Kiado, Budapest 1970.
39. W. Sierpinski, Une theoreme generale sur les familles d'ensemble, Fund. Math. 12 (1928), 206-211.
40. R. Sikorski, Boolean Algebras, Springer-Verlag, Berlin, Gottingen, Heidelberg, New York, First edition 1960; Second edition 1964.
41. M. H. Stone, Postulates for Boolean Algebras and Generalized Boolean Algebras, Amer. J. Math. 57 (1935), 703-732.
42. M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37-111.
43. M. H. Stone, Algebraic characterizations of special Boolean rings, Fund. Math. 29 (1937), 223-303.
44. T. Traczyk, Wstep do teorii algebr Boole'a, PWN, Warszawa 1970.
45. D. A. Vladimirov, Boolean Algebras, Nauka, Moscow 1969 (in Russian).