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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 | ||
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