The Structure Of Several Classes Of Generalized Loops And Pseudo BCI/CI-algebra

The word Loop has many different meanings in mathematics.In this thesis,Loop comes from the equation root problem in algebra,it refers to a quasi-group with units(a quasi-group is an algebraic system that has solutions for both xa=b and ax=b).This thesis mainly studies several kinds of generalized Loops and their related logical algebras—Pseudo BCI-algebra,which are closely related to fuzzy logic.In 2018,American scholar Florentin Smarandache proposed the neutrosophic triplet group(NETG),which is a generalization of group structure and can be regarded as a generalized Loop.In this paper,the research idea of neutrosophic triplet group is applied to the Abel Grassmann’s groupoid(AG-groupoid),and a new concept of AG-NET-Loop is put forward,its basic properties are studied from the point of view of local unit element and local inverse element,and the structure of AG-(l,l)-Loop,a broader class of generalized Loop,is further explored.Pseudo BCI-algebra was proposed by Professor Dudek and Professor Jun in 2008.It is a non-commutative generalization of the famous nonclassical logical algebra BCK/BCI-algebra,and is closely related to noncommutative fuzzy logical algebra(such as pseudo BL-algebra,pseudo MTL-algebra,non-commutative residuals,etc.).Based on the existing research results of pseudo BCI-algebras,this paper studies the structure of some special pseudo BCI-algebras,and analyzes their internal relations with the aforementioned generalized Loops.The research results are as follows:(1)Some new properties of the neutrosophic triplet group(NETG)are given.It is proved for the first time that the neutrosophic triplet group are equivalent to completely regular semigroups.A new concept of weakly commutative neutrosophic triplet group is presented.It is proved that the weakly commutative neutrosophic triplet group is equivalent to Clifford semigroup.(2)As a special Abel Grassmann’s groupoid(AG-groupoid),a new concept of AG-NET-Loop was proposed,its basic properties were studied,and three important conclusions were proved:Weakly commutative AG-NET-Loop is equivalent to commutative regular semigroup.Any AG-NET-Loop can be decomposed into the unintersected union of maximal AG-subgroups.The concept of AG-NET-Loop is further extended,and the new concept of AG-(l,l)-Loop is introduced.It is proved that the(local)neutral element of each element in the strong AG-(l,l)-Loop is unique.The decomposition theorem of strong AG-(l,l)-Loop is established.(3)The basic structure and properties of AG-group(a kind of generalized Loop)are analyzed.Two new concepts of involution AG-group and generalized involution AG-group are introduced.It is proved that the generalized involution AG-group must be homomorphic to an involution AG-group and an involution AG-group can eventually induce a commutative group.The concept of filter is introduced in AG-group and the quotient structure is established.(4)In pseudo BCI-algebra,the concepts of quasi-maximal element and quasi-left identity element are introduced.Two special pseudo BCI-algebras,QM-pseudo BCI-algebras(each element is a quasi-maximal element)and weakly combined pseudo BCI-algebras(WA-pseudo BCI-algebras),are proposed.Some special properties and structures are given.It is proved that every QM-pseudo BCI-algebra is the KG-union of quasi-alternative BCK-algebra and an antigrouped pseudo BCI-algebra.The structure theorem of WA-pseudo BCI-algebra is given.By introducing the adjoint semigroup of pseudo BCI-algebra,the relation between pseudo BCI-algebra and generalized Loop is analyzed.Any Abel Grassman BCI-algebra is AG-NET-Loop.