Two Classes Of Non-associative Groupoids And Their Structures

The study of semigroups has achieved many meaningful results,including the study of basic properties and structures(such as decomposition,etc.)and the application of semigroup algebra in the fields of computer science,statistics,topology,probability and combination.Semigroups are groupoids satisfying the associative law(mathematically,groupoids have different meanings.In this paper,groupoids are also called group embryo,which are algebraic systems with a binary operation on non-empty sets.),non-associative groupoids are the basic components of complex algebras such as semi-rings,rings,and non-associative algebras(Lie algebra,alternate algebra,etc.).Literature review shows that in the theory of non-associative algebra,non-associative fuzzy logic,decision-making and other theoretical aspects,as well as in image processing,network and other applications,non-associative is of great significance,and has achieved a series of results.In this paper,starting from two non-associative operation laws(Tarski Associative Law,Type-2 Cyclic Associative Law),two types of non-associative groupoids are proposed,which are called TA-Groupoids(Tarski Associative Groupoids)and T2CA-Groupoids(Type-2 Cyclic Associative Groupoids);draw lessons from the research ideas and methods of semigroups and regular semigroups,deeply study some properties(such as cancellativity,direct product,etc.),given the decomposition theorems and equivalent characterizations of several algebraic structures;combining the two types of non-associative groupoids and NET-Groupoids(Neutrosophic Extended Triplet Groupoids),new concepts of TA-NET-Groupoids and T2CA-NET-Groupoids are proposed,from the perspectives of local identity elements and local inverse elements,they have studied their basic properties and structures.Explored the relationship between these two types of groupoids.The main research results obtained in this paper are as follows:(1)Some basic properties of the Tarski Associative Groupoid(TA-groupoid)are obtained,and the relationship between it and other algebraic systems is analyzed;the new concept of TA-NET-Groupoid and weak commutative TA-NET-Groupoid(WC-TA-NET-Groupoid)was introduced for the first time,which proved the following conclusions:TA-NET-Groupoid is equivalent to WC-TA-NET-Groupoid,the local identity element of each element of TA-NET-Groupoid is unique;finally,the decomposition theorem of TA-NET-Groupoid is given:TA-NET-Groupoid can be decomposed into disjoint unions of maximal subgroups.(2)Starting from another form of CA(Cyclic Associative)law,a new concept of T2CA-Groupoid(Type-2 Cyclic Associative-Groupoid)is proposed,and a series of properties of T2CA-Groupoid are given;the concept of T2CA-NET-Groupoid is proposed,which proves that T2CA-NET-Groupoid is equivalent to commutative regular semigroups;as the promotion of T2CA-NET-Groupoid,T2CA-(1,1)-NET-Groupoid,T2CA-(1,r)-NET-Groupoid,T2CA-(r,r)-NET-Groupoid,T2CA-(r,1)-Groupoid are introduced,proved that they are equivalent to commutative regular semigroups;the introduction of the new concept of QNET-Groupoids(Quasi Neutrosophic Extended Triplet Groupoids)provides sufficient and necessary conditions for groupoids to become T2CA-QNET-Groupoids,T2CA-NET-Groupoids and CA-NET-Groupoids.(3)The concepts of regular CA-Groupoid and inverse CA-Groupoid are introduced,and it is proved for the first time that both regular CA-Groupoid and inverse CA-Groupoid are equivalent to CA-NET-Groupoid;The H class of the regular CA-Groupoid is a group;finally,with the help of regular CA-Groupoids,the relationship between the two types of non-associative Groupoids(TA-Groupoid s,T2CA-Groupoid s)proposed in this paper is studied:a.The interchangeable semigroup is both TA-Groupoid and T2CA-Groupoid.b.TA-Groupoid and T2CA-Groupoid do not contain each other.c.The left-changeable TA-Groupoid is T2CA-Groupoid,and the left-changeable T2CA-Groupoid is TA-Groupoid.