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The existence of states based on Glivenko semihoops

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Abstract

In this paper, we mainly investigate the existence of states based on the Glivenko theorem in bounded semihoops, which are building blocks for the algebraic semantics for relevant fuzzy logics. First, we extend algebraic formulations of the Glivenko theorem to bounded semihoops and give some characterizations of Glivenko semihoops and regular semihoops. The category of regular semihoops is a reflective subcategory of the category of Glivenko semihoops. Moreover, by means of the negative translation term, we characterize the Glivenko variety. Then we show that the regular semihoop of regular elements of a free algebra in the variety of Glivenko semihoops is free in the corresponding variety of regular semihoops. Similar results are derived for the semihoop of dense elements of free Glivenko semihoops. Finally, we give a purely algebraic method to check the existence of states on Glivenko semihoops. In particular, we prove that a bounded semihoop has Bosbach states if and only if it has a divisible filter, and a bounded semihoop has Riečan states if and only if it has a semi-divisible filter.

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Acknowledgements

We would like to thank the anonymous referee for valuable comments and suggestions for improving the paper. This work is supported by the National Natural Science Foundation of China (12171294, 12001423, 11901371, 11871320, 11971286) and the Fundamental Research Funds for the Central Universities (GK202003003, GK202101009).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Pengfei He

  2. School of Science, Xi’an Shiyou University, Xi’an, 710065, China

    Juntao Wang

  3. College of Mathematics, Northwest University, Xi’an, 710127, China

    Jiang Yang

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  1. Pengfei He
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  2. Juntao Wang
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  3. Jiang Yang
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Correspondence to Jiang Yang.

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He, P., Wang, J. & Yang, J. The existence of states based on Glivenko semihoops. Arch. Math. Logic 61, 1145–1170 (2022). https://doi.org/10.1007/s00153-022-00830-w

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