## Abstract

The main goal of this paper is to investigate monadic bounded hoops and prove the completeness of the monadic hoop logic. In the paper, we introduce monadic bounded hoops: A variety of bounded hoops equipped with universal and existential quantifiers. Also, we study some properties of them and obtain some conditions under which a bounded hoop becomes a Heyting algebra and BL-algebra. In addition, we introduce and investigate monadic filters in monadic bounded hoops. Using monadic filters on monadic bounded hoops, we characterize simple monadic bounded hoops. Moreover, we focus on algebraic structures of the set \(\textit{MF}[H]\) of all monadic filters on monadic bounded hoops and obtain that \(\textit{MF}[H]\) forms a complete Heyting algebra. Further, we discuss relations between monadic bounded hoops and some related algebraic structures, likeness other monadic algebras, bounded hoops with regular Galois connection and rough approximation spaces. Finally, as an application of monadic bounded hoops, we prove the completeness of monadic hoop logic. These results will provide a more general algebraic foundations of soft computing intended as a method for dealing with uncertain information, fuzzy information and decision making.

## Keywords

Logical algebra Hoop Quantifier Monadic filter Monadic hoop logic## Notes

### Acknowledgements

The authors are extremely grateful to the editor and the referees for their valuable comments and helpful suggestions which help to improve the presentation of this paper. This study was funded by a grant of National Natural Science Foundation of China (11571281, 11601302), Postdoctoral Science Foundation of China (2016M602761) and the Fundamental Research Funds for the Central Universities (GK201603004).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Human participants or animals

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

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