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States on wEMV-algebras

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Abstract

Recently in Dvurečenskij 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.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

  2. Department of Algebra and Geometry, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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Anatolij Dvurečenskij: I am declaring that I have no conflicts of interest.

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Dedicated to the memory of Prof. Domenico Candeloro, an outstanding scholar and a friend.

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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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Dvurečenskij, A. States on wEMV-algebras. Boll Unione Mat Ital 13, 515–527 (2020). https://doi.org/10.1007/s40574-020-00233-w

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