QM-BZ-Algebras and Quasi-Hyper BZ-Algebras

Abstract

BZ-algebra, as the common generalization of BCI-algebra and BCC-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-BZ-algebra and quasi-hyper BZ-algebra are proposed and their structures and constructions are studied. First, the definition of QM-BZ-algebra is presented, and the structure of QM-BZ-algebra is obtained: Each QM-BZ-algebra is KG-union of quasi-alter BCK-algebra and anti-grouped BZ-algebra. Second, the new concepts of generalized quasi-left alter (hyper) BZ-algebras and QM-hyper BZ-algebra are introduced, and some characterizations of them are investigated. Third, the definition of quasi-hyper BZ-algebra is proposed, and the relationships among BZ-algebra, hyper BZ-algebra, quasi-hyper BCI-algebra, and quasi-hyper BZ-algebra are discussed. Finally, several special classes of quasi-hyper BZ-algebras are studied in depth and the following important results are proved: (1) an anti-grouped quasi-hyper BZ-algebra is an anti-grouped BZ-algebra; (2) every generalized anti-grouped quasi-hyper BZ-algebra corresponds to a semihypergroup.