In this paper, we study the connection between generalized quasi-left alter BCI-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-BCI algebra, in which every element is a quasi-minimal element, and prove that each QM-BCI algebra is equivalent to generalized quasi-left alter BCI-algebra. Then, we introduce the notion of generalized quasi-left alter-hyper BCI-algebra and prove that every generalized quasi-left alter-hyper BCI-algebra is a generalized quasi-left alter BCI-algebra. Next, we propose a new notion of quasi-hyper BCI algebra and discuss the relationship among them. Moreover, we study the subalgebras of quasi-hyper BCI algebra and the relationships between Hv-group and quasi-hyper BCI-algebra, hypergroup and quasi-hyper BCI-algebra. Finally, we propose the concept of a generalized quasi-left alter quasi-hyper BCI algebra and QM-quasi hyper BCI-algebra and discuss the relationships between them and related BCI-algebra.

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.

Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the elements in the regularity conditions, some new regularity conditions (collectively referred to as transposition regularity) are obtained, and the concepts of various transposition regular semigroups are introduced (L1/L2/L3, R1/R2/R3-transposition regular semigroups, etc.). Their relations with completely regular semigroups and left (right) regular semigroups, proposed by Clifford and Preston, are analyzed. Their properties and structures are studied from the aspects of idempotents, local identity elements, local inverse elements, subsemigroups and so on. Their decomposition theorems are proved respectively, and some new necessary and sufficient conditions for semigroups to become completely regular semigroups are obtained.

In this paper, we introduce the new concept of a hyper BZ-algebra which is a generalization of BZ-algebra and hyper BCI-algebra, and give some examples and basic properties. We discuss the relationships among hyper BZ-algebras, hyper BCC-algebras and hyper BCIalgebra. Moreover, we propose the concepts of antigrouped hyper BZ-algebras and generalized anti-grouped hyper BZ-algebras, and prove that the following important results:(1) Every anti-grouped hyper BZ-algebra is an antigrouped BZ-algebra;(2) Every generalized anti-grouped hyper BZ-algebra corresponds to a semihypergroup. Finally, we present a method to construct a new hyper BZ-algebra by using a hyper BCC-algebra and a standard generalized anti-grouped hyper BZ-algebra.

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group.