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Generalized Bosbach and Riečan states on nucleus-based-Glivenko residuated lattices

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Abstract

Bosbach and Riečan states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Riečan states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mureşan through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the present paper, the Glivenko theorem is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Riečan state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Bin Zhao & Hongjun Zhou

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  1. Bin Zhao
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  2. Hongjun Zhou
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Correspondence to Hongjun Zhou.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 61005046 and 11171196), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20100202120012) and the Natural Science Program for Basic Research of Shaanxi Province, China (Grant no. 2010JQ8020).

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Zhao, B., Zhou, H. Generalized Bosbach and Riečan states on nucleus-based-Glivenko residuated lattices. Arch. Math. Logic 52, 689–706 (2013). https://doi.org/10.1007/s00153-013-0338-7

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