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(2201-7188) States on weak pseudo EMV-algebras. II. Representations of states | ||
| Iranian Journal of Fuzzy Systems | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 05 اردیبهشت 1401 اصل مقاله (185.44 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2022.6891 | ||
| نویسنده | ||
A. Dvurecenskij 
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| Institute of Mathematics, Slovak Academy of Sciences, Stefanikova 49, SK-814 73 Bratislava, Slovakia | ||
| چکیده | ||
| Recently in [7, 8], 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 de ne a state as a nitely additive mapping from a wPEMValgebra 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 wPEMVhomomorphisms 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 σ-additive probability measure. | ||
| کلیدواژه‌ها | ||
| Pseudo MV-algebra؛ pseudo EMV-algebra؛ wPEMV-algebra؛ generalized Boolean algebra؛ state؛ statemorphism؛ extremal state؛ pre-state؛ maximal and normal ideal؛ weak convergence؛ simplex؛ integral representation | ||
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