Browse > Article
http://dx.doi.org/10.5392/IJoC.2011.7.4.056

Logic of Quantum Mechanics for Information Technology Field  

Yon, Yong-Ho (Innovation Center for Engineering Education, Mokwon University)
Publication Information
International Journal of Contents / v.7, no.4, 2011 , pp. 56-63 More about this Journal
Abstract
Quantum mechanics is a branch of physics for a mathematical description of the particle wave, and it is applied to information technology such as quantum computer, quantum information, quantum network and quantum cryptography, etc. In 1936, Garrett Birkhoff and John von Neumann introduced the logic of quantum mechanics (quantum logic) in order to investigate projections on a Hilbert space. As another type of quantum logic, orthomodular implication algebra was introduced by Chajda et al. This algebra has the logical implication as a binary operation. In pure mathematics, there are many algebras such as Hilbert algebras, implicative models, implication algebras and dual BCK-algebras (DBCK-algebras), which have the logical implication as a binary operation. In this paper, we introduce the definitions and some properties of those algebras and clarify the relations between those algebras. Also, we define the implicative poset which is a generalization of orthomodular implication algebras and DBCK-algebras, and research properties of this algebraic structure.
Keywords
Hilbert algebras; implication algebras; DBCK-algebras; orthomodular implication algebras; implicative posets;
Citations & Related Records
연도 인용수 순위
  • Reference
1 K. Iseki, "An algebra related with a propositional calculus," Proc. Japan Acad., Vol. 42, No. 1, 1966, pp. 26-29.   DOI
2 K. Iseki and S. Tanaka, "An introduction to the theory of BCK-algebras," Math. Jpn., Vol. 23, 1978, pp. 1-26.
3 L. Henkin, "An algebraic characterization of quantifiers," Fund. Math., Vol. 37, 1950, pp. 63-74.   DOI
4 E. L. Marsden, Compatible elements in implication models," J. Philos. Logic, Vol. 1, 1972, pp. 156-161.   DOI
5 E. L. Marsden, "A note on implicative models," Notre Dame J. Form. Log., Vol. XIV, No. 1, 1973, pp. 139-144.
6 D. Busneag, "Hertz algebras of fractions and maximal Hertz algebra of quotients," Math. Jpn., Vol. 39, No. 3, 1993, pp. 461-469.
7 A. Diego, "Sur les algebres de Hilbert," Collection de Logique Mathematique, Ser. A, Vol 21, 1966.
8 A. V. Figallo, G. Ramon, and S. Saad, "A note on the Hilbert algebras with infimum," Mat. Contemp., Vol. 24, 2003, pp. 23-37.
9 R. Halas, "Remarks on commutative Hilbert algebras," Math. Bohem., Vol. 127, No. 4, 2002, pp. 525-529.
10 W. A. Dudek, "On embedding Hilbert algebras in BCKalgebras," Math. Morav., Vol. 3, 1999, pp. 25-29.
11 J. C. Abbott, "Algebras of implication and semi-lattices," Seminarire Dubreil (Algebre et theorie des nombres), Vol. 20e, No. 2, 1966-1967, exp. n^0 20, pp. 1-8.
12 J. Meng, "Implication algebras are dual to implicative BCK-algebras," Soochow J. Math., Vol. 22, No. 4, 1996, pp. 567-571.
13 G. Birkhoff and J. von Neumann, "The logic of quantum mechanics," Ann. of Math., Vol. 37, 1936, pp. 822-843.
14 K. Husimi, "Studies on the foundations of quantum mechanics I," Proc. of the physicomath. Soc. of Japan, Vol. 19, 1937, pp. 766-789.
15 G. Kalmbach, Orthomodular lattices, Academic Press, New York, 1983.
16 P. D. Finch, "On the lattice structure of quantum logic," Bull. Austral. Math. Soc., Vol. 1, 1969, pp. 333-340.   DOI
17 K. H. Kim and Y. H. Yon, "Dual BCK-algebra and MValgebra," Sci. Math. Jpn., Vol. 66, No. 2, 2007, pp. 247-253.
18 P. D. Finch, "Quantum logic as an implication algebra," Bull. Austral. Math. Soc., Vol. 2, 1970, pp. 101-106.   DOI
19 I. Chajda, R. Halas, and H. Langer, "Orthomodular implication algebras," Int. J. of Theor. Phys., Vol. 40, No. 11, Nov. 2001, pp. 1875-1884.   DOI   ScienceOn
20 R. A. Borzooei and S. K. Shoar, "Implication algebras are equivalent to the dual implicative BCK-algebras," Sci. Math. Jpn., Vol. 63, No. 3, 2006, pp. 429-431.
21 A. Walendziak, "On Commutative BE-algebras," Sci. Math. Jpn., Vol. 69, No. 2, 2009, pp. 281-284.