On the relationships between hybrid generalized Bosbach states and L-filters in non-commutative residuated lattices

  • Weibing ZuoEmail author


States and filters play the essential roles in studying logical algebras. In this paper, we study the relationships between hybrid generalized Bosbach states and L-filters and hybrid L-filters in non-commutative residuated lattices. Particularly, two types of L-filters and hybrid L-filters and their subclasses are defined, and some of their properties are obtained. Then, relationships between special types of (hybrid) L-filters and the hybrid generalized Bosbach states are considered where hybrid generalized Bosbach states are characterized by some type I or type II (hybrid) L-filters with additional conditions. Associated with these relationships, new subclasses of hybrid generalized Bosbach states such as implicative type IV-1, IV-2, V-1, V-2, VI-1, VI-2 states, Involution type IV-1, IV-2 states and Boolean type IV-1, IV-2 states are introduced, and the relationships between various types of hybrid generalized Bosbach states are investigated in detail.


Non-commutative residuated lattice Hybrid L-filter Hybrid generalized Bosbach state Implicative type IV-1(resp. IV-2, V-1, V-2, VI-1, VI-2) state 



This work was supported by the Natural Science Foundation of Henan Province of China (No. 152300410112).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.

Human and animal participants

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsNorth China University of Water Resources and Electric PowerZhengzhouChina

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