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g-States on unital weak pseudo EMV-algebras

  • Foundation, algebraic, and analytical methods in soft computing
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Abstract

Recently in Dvurečenskij and Zahiri (J Appl Log IfCoLog J Log Appl 8:2365–2399, 2021b, J Appl Log IfCoLog J Log Appl 8:2401–2433, 2021c), new algebras, called weak pseudo EMV-algebras (wPEMV-algebras in short), were introduced. The authors do not assume the existence of a top element —they generalize MV-algebras, pseudo MV-algebras, and pseudo EMV-algebras. A g-state is defined on a unital wPEMV-algebra M as a mapping from M into the positive half-line of reals such that it preserves a partial addition \(+\), and in a fixed strong unit, it takes the value 1. They form a Bauer simplex, and extremal points are exactly g-states whose kernel is a maximal and normal ideal. We show that extremal g-states generate all g-states, and it can happen that in some unital wPEMV-algebra, even commutative, there is no g-state. We present some conditions for existence of g-states and establish an integral representation of g-states. In addition, we give a topological characterization of the spaces of g-states and extremal g-states, respectively. Moreover, discrete g-states are investigated.

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Acknowledgements

The author is very indebted to an anonymous referee for his/her remarks which improved the paper’s readability.

Funding

The study was funded by the grant of the Slovak Research and Development Agency under contract APVV-20-0069 and by the Grant VEGA No. 2/0142/20 SAV.

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Authors and Affiliations

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

    Anatolij Dvurečenskij

  2. Department of algebraic geometry, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Dvurečenskij, A. g-States on unital weak pseudo EMV-algebras. Soft Comput 27, 4353–4368 (2023). https://doi.org/10.1007/s00500-023-07850-5

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