Abstract
In the present paper, a new variation of a nonnegative real-valued function \(\rho \) defined on a subfamily of a quantum logic P is proposed, and the notion of tightness of \(\rho \) is studied. Various crucial results are proved, and subsequently we have obtained extensions, viz. f-extension, \(\delta \)-extension and \(\sigma \)-extension of a tight function \(\rho ;\) continuity of extensions of \(\rho \) with respect to an approximating family in P is discussed.
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References
Adamski W (1984) Extensions of tight set functions with applications in topological measure theory. Trans Am Math Soc 283:353–368
Adamski W (1987) On regular extensions of contents and measures. J Math Anal Appl 127:211–225
Avallone A, De Simone A (2001) Extensions of modular functions on orthomodular lattices. Ital J Pure Appl Math Soc 9:109–122
Beaver OR, Cook TA (1977) States on quantum logics and their connection with a theorem of Alexandroff. Proc Am Math Soc 67:133–134
Beltrametti EG, Cassinelli G (1981) The logic of quantum mechanics. Addison-Wesley, Reading
Beran L (1984) Orthomodular lattices, algebraic approach. D. Reidel, Holland
Birkhoff G, Von Neumann J (1936) The logic of quantum mechanics. Ann Math 37:823–834
Butnariu D, Klement P (1993) Triangular norm-based measures and games with fuzzy coalitions. Kluwer, Dordrecht
Bonzio S, Chajda I (2017) A note on orthomodular lattices. Int J Theor Phys 56:3740–3743
Dumitrescu D (1993) Fuzzy measures and the entropy of fuzzy partitions. J Math Anal Appl 176:359–373
Dvurečenskij A (2017) On orders of observables on effect algebras. Int J Theor Phys 56:4112–4125
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publishers, Dordrecht
Kagan E, Ben-gal I (2011) Navigation of quantum-controlled mobile robots. In: Topalov A (ed) Recent advances in mobile robotics. InTech, Rijeka, pp 311–326
Kalmbach G (1983) Orthomodular lattices. Academic Press, London
Keimel K, Lawson JD (2005) Measure extension theorems for \(T_0\)-spaces. Top Appl 149:57–83
Khare M, Gupta S (2010) Non-additive measures, envelops and extensions of quasi-measures. Sarajevo J Math 6(18):35–49
Khare M, Gupta S (2008) Extension of non-additive measures on locally complete \(\sigma \)-continuous lattices. Novi Sad J Math 38(2):15–23
Khare M, Roy S (2008a) Conditional entropy and the Rokhlin metric on an orthomodular lattice with Bayesian state. Int J Theor Phys 47(5):1386–1396
Khare M, Roy S (2008b) Entropy of quantum dynamical systems and sufficient families in orthomodular lattices with Bayesian state. Commun Theor Phys 50:551–556
Khare M, Singh AK (2008) Weakly tight functions, their Jordan type decomposition and total variation in effect algebras. J Math Anal Appl 344(1):535–545
Khare M, Singh B, Shukla A (2018) Approximation in quantum measure spaces. Math Slovaca 68(3):491–500
Kelley JL, Nayak MK, Srinivasan TP (1972) Premeasure on lattices of sets II. In: Symposium on vector measures, Salt Lake City, Utah
Marczewski E (1953) On compact measures. Fund Math 40:113–124
Markechová D (1993) The entropy of complete fuzzy partitions. Math Slovaca 43:1–10
Ming LY, Kang LM (1997) Fuzzy topology. World Scientific Publ. Co., Singapore
Morales P (1981) Extension of a tight set function with values in a uniform semigroup. In: Kölzow D, Maharam-Stone D (eds) Measure theory, Oberwolfach 1981, Lecture Notes in Mathmatics, vol 45. Springer, pp 282–292
Nayak MK, Srinivasan TP (1975) Scalar and vector valued premeasures. Proc Am Math Soc 48(2):391–396
Pap E (1995) Null-additive set functions. Kluwer Academic Publishers, Dordrecht
Pták P, Pulmannová S (1991) Orthomodular structures as quantum logics. Kluwer, Dordrecht
Riečan B (1979) The measure extension theorem for subadditive probability measures in orthmodular \(\sigma \)-continuous lattices. Commun Math Univ Carol 202:309–315
Topsøe F (1970a) Compactness in spaces of measures. Stud Math 36:195–212
Topsøe F (1970b) Topology and measure. In: Dold A, Eckmann B (eds) Lecture notes in mathmatics, vol 133. Springer
Varadarajan VS (1968) Geometry of quantum theory, vol 1. Van Nostrand, Princeton
Yali W, Yichuan Y (2017) Notes on quantum logics and involutive bounded posets. Soft Comput 21:2513–2519
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The authors are grateful to the anonymous referees for their valuable suggestions toward the improvement of the paper.
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The second author acknowledges with gratitude the financial support by Department of Science and Technology (DST), New Delhi, India, under INSPIRE fellowship No. IF160721.
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Khare, M., Pandey, P. Extensions of a tight function and their continuity in quantum logic. Soft Comput 23, 11389–11398 (2019). https://doi.org/10.1007/s00500-019-04057-5
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DOI: https://doi.org/10.1007/s00500-019-04057-5