Monadic bounded hoops

• Published in: Soft computing (Berlin, Germany) Vol. 22; no. 6; pp. 1749 - 1762
• Main Authors: Wang, Juntao, Xin, Xiaolong, He, Pengfei
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2018; Springer Nature B.V
• Subjects: Algebra; Artificial Intelligence; Boolean; Completeness; Computational Intelligence; Computer science; Control; Engineering; Foundations; Fuzzy logic; Hoops; Information processing; Mathematical Logic and Foundations; Mechatronics; Robotics; Semantics; Soft computing; Hoop; Quantifier; Monadic filter; Logical algebra; Monadic hoop logic
• ISSN: 1432-7643; 1433-7479
• DOI: 10.1007/s00500-017-2648-x

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 MF [ H ] of all monadic filters on monadic bounded hoops and obtain that 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.