States on wEMV-algebras
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States on wEMV-algebras Anatolij Dvureˇcenskij1,2 Received: 5 May 2020 / Accepted: 29 May 2020 © Unione Matematica Italiana 2020
Abstract Recently in Dvureˇcenskij and Zahiri (A variety containing EMV-algebras and Pierce sheaves, arXiv:1911.06625), new algebras called wEMV-algebras, which generalize MV-algebras, generalized Boolean algebras and EMV-algebras, were founded, and for these algebras a top element is not assumed a priori. For this class we define a state as a mapping from a wEMV-algebra into the real interval [0, 1] which preserves a kind of subtraction of two comparable elements and attaining the value 1 in some element. It can happen that some wEMV-algebras are stateless, e.g. cancellative ones. We characterize extremal states just as state-morphisms which are wEMV-homomorphisms from an algebra into the real interval [0, 1]. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal ideals having a special property. Moreover, we prove that under some conditions every state on a wEMV-algebra is a weak limit of a net of convex combinations of state-morphisms. Keywords EMV-algebra · Wemv-algebra · MV-algebra · Generalized Boolean algebra · State · State-morphism · Extremal state · Pre-state Mathematics Subject Classification 03G12 · 03B50 · 06C15 · 81P15
1 Introduction States as finitely additive mappings with values in the real interval [0, 1] attaining the value 1 in a top element are of primary notions of quantum structures. They are important for mathematical foundations of quantum mechanics and states are appearing in many important
Dedicated to the memory of Prof. Domenico Candeloro, an outstanding scholar and a friend. The paper has been supported by the Grant of the Slovak Research and Development Agency under contract APVV-16-0073 and the Grant VEGA No. 2/0142/20 SAV.
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Anatolij Dvureˇcenskij [email protected]
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Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
2
Department of Algebra and Geometry, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
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A. Dvureˇcenskij
quantum structures like quantum logics, orthomodular lattices or posets, effect algebras, etc. For more information about quantum structures and states on them we recommend to see e.g. [6,7]. In the last three decades states entered also in pure algebras like MV-algebras, see [15], or BL-algebras, see [11]. States were studied also on non-commutative algebras like pseudo MV-algebras [5]. It was recognized that there is a nice characterization of extremal states as state-morphisms, that is homomorphisms into the real interval [0, 1]. Moreover, there is a oneto-one correspondence between state-morphisms and maximal ideals. Introducing the weak topology of states it was possible to show that the set of state-morphisms is homeomorphic with the hull-kernel topology of maximal states. In addition, every state is a weak limit of a net of convex combinations of state-morphisms. In [14,17], it was shown that every
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