Soft Computing

, Volume 23, Issue 3, pp 947–959 | Cite as

New types of generalized Bosbach states on non-commutative residuated lattices

  • Weibing ZuoEmail author
Methodologies and Application


Generalized Bosbach states of types I and II and hybrid generalized Bosbach states of types I and II are useful for the development of algebraic theory of probabilistic models for non-commutative fuzzy logics. In this paper, eight types of hybrid generalized Bosbach states of types III-1, III-2, IV-1, IV-2, V-1, V-2, VI-1 and VI-2 (or simply, hybrid types III-1, III-2, IV-1, IV-2, V-1, V-2, VI-1 and VI-2 state) on non-commutative residuated lattices are introduced. The relationships among hybrid generalized Bosbach states and properties of them are studied. Particularly, hybrid type III-1 state (resp. III-2 ) implies type I state (resp. hybrid type I state); hybrid type IV-1 (resp. IV-2) states are a new type of generalized Bosbach state which are different from type I, II and hybrid I, II states; hybrid type V-1 (resp. V-2) states can be equivalently defined by both type I (resp. hybrid type I) states and hybrid type IV-1 (resp. hybrid type IV-2) states; etc. The relationships between new types of generalized Bosbach states and (hybrid) generalized state-morphisms and (hybrid) generalized Riečan states are investigated.


Non-commutative residuated lattice Generalized Bosbach state Hybrid generalized Bosbach state of type III-1 (resp. III-2, IV-1, IV-2, V-1, V-2, VI-1 and VI-2) 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

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

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