Abstract
Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established.
Highlights
An algebraic structure is called a groupoid, if it is well-defined regarding an operation on it.A groupoid satisfying the “cyclic associative law” (that is, x(yz) = y(zx)) is called a cyclic associative groupoid, or CA-groupoid [1,2].as early as 1946, when Byrne [3] studied axiomatization of Boolean algebra, he mentioned the following operation law:z =x
CA-groupoids with unit element degenerate into commutative monoids, and a CA- groupoid with quasi right unit element maybe not a semigroup
The content of this paper as follows: in Section 2, we introduce some basic concepts and cancellative properties on semigroup and AG-groupoid; in Section 3, we give the definitions of cancellative CA-groupoids, left cancellative CA-groupoids, right cancellative CA-groupoids and weak cancellative CA-groupoids, and discuss the relationships about them; in Section 4, we give the definitions of several quasi-cancellative CA-groupoids and power cancellative CA-groupoids, and analyze the relationships about several types cancellative CA-groupoids; in Section 5, we propose the new notion of variant CA-groupoid and some interesting examples, we prove the structure theorem and construction method of variant CA-groupoids
Summary
An algebraic structure is called a groupoid, if it is well-defined regarding an operation on it. In 1954, Sholander [4] mentioned Byrne’s paper [3], and used the term of “cyclic associative law” to express the operation law: (ab)c = (bc)a. CA-groupoids with unit element degenerate into commutative monoids, and a CA- groupoid with quasi right unit element (i.e., there exists e, if x , e, xe = x; and ee , e) maybe not a semigroup The content of this paper as follows: in Section 2, we introduce some basic concepts and cancellative properties on semigroup and AG-groupoid; in Section 3, we give the definitions of cancellative CA-groupoids, left cancellative CA-groupoids, right cancellative CA-groupoids and weak cancellative CA-groupoids, and discuss the relationships about them; in Section 4, we give the definitions of several quasi-cancellative CA-groupoids and power cancellative CA-groupoids, and analyze the relationships about several types cancellative CA-groupoids; in Section 5, we propose the new notion of variant CA-groupoid and some interesting examples, we prove the structure theorem and construction method of variant CA-groupoids
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