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States on weak pseudo EMV-algebras. II. Representations of states | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 3، دوره 19، شماره 4، مهر و آبان 2022، صفحه 17-26 اصل مقاله (186.74 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2022.7083 | ||
| نویسنده | ||
| A. Dvureˇcenskij* | ||
| Institute of Mathematics, Slovak Academy of Sciences, Stef´anikova 49, SK-814 73 Bratislava, Slovakia ˇ | ||
| 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 | ||
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[1] E. M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, Berlin, 1971.
[2] A. Dvurečenskij, States on wEMV-algebras, Bollettino dell’Unione Matematica Italiana, 13 (2020), 515-527.
[3] A. Dvurečenskij, States on weak pseudo EMV-algebras. I. States and states morphisms, Iranian Journal of Fuzzy Systems, 19(4) (2022), 1-15.
[4] A. Dvurečenskij, O. Zahiri, States on EMV-algebras, Soft Computing, 23 (2019), 7513-7536.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] K. R. Goodearl, Partially ordered Abelian groups with interpolation, Mathematical Surveys and Monographs, No. 20, American Mathematics Society, Providence, Rhode Island, 1986.
[10] J. L. Kelley, General topology, Van Nostrand, Priceton, New Jersey, 1955.
[11] T. Kroupa, Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems, 157 (2006), 2771-2782.
[12] G. Panti, Invariant measures in free MV-algebras, Communications in Algebra, 36 (2008), 2849-2861. | ||
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