اخبار و اعلانات
آمار
| تعداد نشریات | 29 |
| تعداد شماره‌ها | 644 |
| تعداد مقالات | 6,315 |
| تعداد مشاهده مقاله | 9,882,366 |
| تعداد دریافت فایل اصل مقاله | 6,471,829 |
States on weak pseudo EMV-algebras. I. States and states morphisms} | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 2، دوره 19، شماره 4، مهر و آبان 2022، صفحه 1-15 اصل مقاله (214.47 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2022.7082 | ||
| نویسنده | ||
| A. Dvureˇcenskij* | ||
| Institute of Mathematics, Slovak Academy of Sciences, \v Stef\'anikova 49, SK-814 73 Bratislava, Slovakia} | ||
| چکیده | ||
| Recently in \cite{DvZa5,DvZa6}, new algebras, called weak pseudo EMV-algebras, wPEMV-algebras for short, were introduced generalizing pseudo MV-algebras, generalized Boolean algebras and pseudo EMV-algebras. For these algebras a top element is not assumed a priori. For this class of algebras, we define a state as a finitely additive mapping from a wPEMV-algebra into the real interval $[0,1]$ which preserves a partial addition of two non-interactive elements and attaining the value $1$ in some element. It can happen that some commutative wPEMV-algebras are stateless, e.g. cancellative ones.\\ % The paper is divided into two parts. Part I deals with basic properties of states and state-morphisms which are wPEMV-homomorphisms from a wPEMV-algebra into the real interval $[0,1]$ endowed as a commutative wPEMV-algebra. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal and normal ideals having a special property. In Part II, we present an analogue of the Krein-Mil'man theorem applied to the set of states. We characterize the space of the state-morphisms of a wPEMV-algebra without top element as a Hausdorff locally compact space in the weak topology of states and we present its Alexandroff's one-point compactification. Moreover, we give an integral representation of any (finitely additive) state by a unique regular Borel $\sigma$-additive probability measure. | ||
| کلیدواژه‌ها | ||
| Pseudo MV-algebra؛ pseudo EMV-algebra؛ wPEMV-algebra؛ generalized Boolean algebra؛ state؛ state-morphism؛ extremal state؛ pre-state؛ maximal and normal ideal؛ weak convergence؛ simplex؛ integral representation | ||
| مراجع | ||
|
[1] K. P. S. Bhashkara Rao, M. Bhashkara Rao, Theory of charges: A study of finitely additive measures, Academic Press, London, New York, 1983.
[2] B. de Finetti, Sul significato soggettivo della probabilitá, Fundamenta Mathematicae, 17 (1931), 298-329. Translated into English as On the subjective meaning of probability, in: P. Monari, D. Cocchi (Eds.), Probabilitáe Induzione, Clueb, Bologna, (1993), 291-321.
[3] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL-algebras: Part I, Multi-Valued Logic, 8 (2002), 673-714.
[4] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL-algebras: Part II, Multi-Valued Logic, 8 (2002), 715-750.
[5] L. E. Dubins, L. J. Savage, How to gamble if you must: Inequalities for stochastic processes, McGraw-Hill, London, 1965.
[6] A. Dvurečenskij, States on pseudo MV-algebras, Studia Logica, 68 (2001), 301-327.
[7] A. Dvurečenskij, Pseudo MV-algebras are intervals in ℓ-groups, Journal of the Australian Mathematical Society, 72 (2002), 427-445.
[8] A. Dvurečenskij, Measures on quantum structures, In: Handbook of Measure Theory, E. Pap (Editor), Elsevier Science, Amsterdam, II (2002), 827-868.
[9] A. Dvurečenskij, States on wEMV-algebras, Bollettino dell’Unione Matematica Italiana, 13 (2020), 515-527. DOI: 10.1007/s40574-020-00233-w.
[10] A. Dvurečenskij, States on weak pseudo EMV-algebras. II. Representations of states, Iranian Journal of Fuzzy Systems, 19(4) (2022), 17-26.
[11] A. Dvurečenskij, S. Pulmannová, New trends in quantum structures, Kluwer Academic Publ., Dordrecht, Ister Science, Bratislava, 2000, 541 + xvi pp.
[12] A. Dvurečenskij, O. Zahiri, On EMV-algebras, Fuzzy Sets and Systems, 373 (2019), 116-148.
[13] A. Dvurečenskij, O. Zahiri, States on EMV-algebras, Soft Computing, 23 (2019), 7513-7536.
[14] A. Dvurečenskij, O. Zahiri, Pseudo EMV-algebras. I. Basic properties, Journal of Applied Logics–IFCoLog Journal of Logics and their Applications, 6 (2019), 1285-1327.
[15] A. Dvurečenskij, O. Zahiri, Pseudo EMV-algebras. II. Representation and states, Journal of Applied Logics– IFCoLog Journal of Logics and their Applications, 6 (2019), 1329-1372.
[16] A. Dvurečenskij, O. Zahiri, A variety containing EMV-algebras and Pierce sheaves, Fuzzy Sets and Systems, 418 (2021), 101-125.
[17] A. Dvurečenskij, O. Zahiri, Weak pseudo EMV-algebras. I. Basic properties, Journal of Applied Logics– IfCoLog Journal of Logics and their Applications, 8 (2021), 2365-2399.
[18] A. Dvurečenskij, O. Zahiri, Weak pseudo EMV-algebras. II. Representation and subvarieties, Journal of Applied Logics– IfCoLog Journal of Logics and their Applications, 8 (2021), 2401-2433.
[19] T. Flaminio, F. Montagna, MV-algebras with internal states and probabilistic fuzzy logics, International Journal of Approximate Reasoning, 50 (2009), 138-152.
[20] N. Galatos, C. Tsinakis, Generalized MV-algebras, Journal of Algebra, 283 (2005), 254-291.
[21] G. Georgescu, Bosbach states on fuzzy structures, Soft Computing, 8 (2004), 217-230.
[22] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras, Multi-Valued Logic, 6 (2001), 95-135.
[23] G. Georgescu, L. Leu¸stean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic Soft Computing, 11 (2005), 153-184.
[24] K. R. Goodearl, Partially ordered Abelian groups with interpolation, Mathematical Surveys and Monographs, No. 20, American Mathematical Society, Providence, Rhode Island, 1986.
[25] P. Hájek, Fuzzy logics with noncommutative conjunctions, Journal of Logic and Computation, 13 (2003), 469-479.
[26] J. L. Kelley, General topology, Van Nostrand, Priceton, New Jersey, 1955.
[27] A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933.
[28] T. Kroupa, Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems, 157 (2006), 2771-2782.
[29] D. Mundici, Interpretation of AF C∗ -algebras in Lukasiewicz sentential calculus, Journal of Functional Analysis, 65 (1986), 15-63.
[30] D. Mundici, Averaging the truth-value in Łukasiewicz logic, Studia Logica, 55 (1995), 113-127.
[31] D. Mundici, Interpretation of de Finetti coherence criterion in Łukasiewicz logic, Annals of Pure Applied Logic, 161 (2009), 235-245.
[32] G. Panti, Invariant measures in free MV-algebras, Communications in Algebra, 36 (2008), 2849-2861.
[33] J. Rach˚unek, A non-commutative generalization of MV-algebras, Czechoslovak Mathematical Journal, 52 (2002), 255-273.
[34] V. S. Varadarajan, Geometry of quantum theory, van Nostrand, Princeton, New Jersey, 1, 1968. | ||
|
آمار تعداد مشاهده مقاله: 369 تعداد دریافت فایل اصل مقاله: 403 |
||