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Relationships between generalized Bosbach states and L-filters on residuated lattices

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Abstract

Generalized Bosbach states and filters on residuated lattices have been extensively studied in the literature. In this paper, relationships between generalized Bosbach states and residuated-lattice-valued filters, also called L-filters, on residuated lattices are investigated. Particularly, type I and type II L-filters and their subclasses are defined, and some their properties are obtained. Then relationships between special types of L-filters and the generalized Bosbach states are considered where generalized Bosbach states are characterized by some type I or type II L-filters with additional conditions. Associated with these relationships, new subclasses of generalized Bosbach states such as implicative type IV, V, VI states, fantastic type IV states and Boolean type IV states are introduced, and the relationships between various types of generalized Bosbach states are investigated in detail. In particular, the existence of several generalized Bosbach states is provided and, as application, some typical subclasses of residuated lattices such as Rl-monoids, Heyting algebras and Boolean algebras are characterized by these generalized Bosbach states.

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Acknowledgments

This research was supported by AMEP (DYSP) of Linyi University (Grant No. LYDX2014BS017), the Natural Science Foundation of Shandong Province (Grant No. ZR2013FL006). The author is very grateful to the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper.

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  1. School of Science, Linyi University, Linyi, 276005, People’s Republic of China

    Zhen Ming Ma & Wei Yang

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  1. Zhen Ming Ma
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  2. Wei Yang
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Correspondence to Zhen Ming Ma.

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Communicated by A. Di Nola.

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Ma, Z.M., Yang, W. Relationships between generalized Bosbach states and L-filters on residuated lattices. Soft Comput 20, 3125–3138 (2016). https://doi.org/10.1007/s00500-015-1939-3

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