Abstract
Inspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.
This work was supported by the grant of National Natural Science Foundation of China (11971384), China 111 Project (B16037), the Fundamental Research Funds for the Central Universities (JB150115), the Shaanxi Innovation Team Project (2018TD-007) and (201809168CX9JC10).
Communicated by Anatolij Dvurečenskij
Acknowledgement
This research is partially supported by a grant of National Natural Science Foundation of China (11971384, 61976130), China 111 Project (B16037), Scientific Research Program Funded by Shanxi Provincial Education Department (18JS042), the Shaanxi Innovation Team Project (2018TD-007).
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