Cyclic Associative Groupoids And Their Structures

This paper is the first time to put forward a new concept of cyclic associa-tive groupoid(CA-groupoid)based on the background of cyclic associative non-associative ring,left weak novikov algebra and CA-AG-groupoid,and gives some examples of CA-groupoid(In this paper,the groupoid is also called "group embryo",which refers to the algebraic system composed of a non-empty set and a binary operation on it),and studied its basic properties.Meanwhile,the concept of CA-NET-groupoid is introduced by combining the NETG with the CA-grou-poid,and the relationship between the CA-NET-groupoid and the related algebra is analyzed.Furthermore,a new concept of CA-semiring is introduced and the related properties and structural characteristics of the CA-semiring and the CA-ring are studied deeply.The main results obtained in this paper are as follows:(1)The basic properties of CA-groupoid are given,the necessary and sufficient condition for a CA-groupoid to have left unit element is that it has right unit element.The relationship between the CA-groupoid,the semigroup and the AG-groupoid is systematically analyzed.The concept of some cancella-tive CA-groupoids are proposed,proved the following conclusions:the CA-groupoid is concluded that the cancellative CA-subgroupoid and a CA-subgrou-poid with non-cancellative element.The CA-band and CA-3-band are proved to be quasi-cancellative.The inclusion relationship between cancellative CA-grou-poid,separative CA-groupoid,quasi-cancellative CA-groupoid and power-cancellative CA-groupoid is systematically analyzed.(2)The definition and the examples of CA-NET-groupoids are given,and proved the uniqueness of local unit element in CA-NET-groupoid.Through the analysis of different local units,decomposition theorem of the CA-NET-grou-poid is obtained.Meanwhile,having put forward the concept of variant CA-groupoid and giving corresponding examples,found some characteristics of the quasi-right unit element in variant CA-groupoids.Further,decomposition theorem of the variant CA-groupoid is proved,and a construction method of the variant CA-groupoid is given(see Theorem 4.9).Through the study of a large number of examples of variant CA-groupoid found by matlab,three different methods of variant CA-groupoid are given.(3)The concepts of CA-semiring and CA-ring are introduced,and the corresponding examples are given,and studied its basic properties.The concepts of CA-subsemiring,homomorphism and isomorphism are given on the CA-semiring,the basic theorem of homomorphism and two isomorphism theorems on the CA-semiring are proved.The concept of ideal of CA-ring is given,and its basic properties are studied,the homorphism basic theorem and isomorphism theorem of the CA-ring are given.