The $n^{th}$ commutativity degree of semigroups
محورهای موضوعی : Group theory
M. Ghaneei
1
(Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
M. Azadi
2
(Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
کلید واژه: commutativity degree, Probability, Quasi-commutative semigroups,
چکیده مقاله :
‎For a given positive integer $n$‎, ‎the $n^{th}$ commutativity degree‎ ‎of a finite non-commutative semigroup $S$ is defined to be the‎ ‎probability of choosing a pair $(x,y)$ for $x‎, ‎y \in S$ such‎ ‎that $x^n$ and $y$ commute in $S$‎. ‎If for every elements $x$ and $y$ of‎ ‎an associative algebraic structure $(S,.)$ there exists a‎ ‎positive integer $r$ such that $xy =y^{r}x$‎, ‎then $S$ is called‎ ‎quasi-commutative‎. ‎Evidently‎, ‎every abelian group or commutative‎ ‎semigroup is quasi-commutative‎. ‎In this paper‎, ‎we study the‎ ‎$n^{th}$ commutativity degree of certain classes of‎ ‎quasi-commutative semigroups‎. ‎We show that the‎ ‎$n^{th}$ commutativity degree of such structures is greater than $\dfrac{1}{2}$‎. ‎Finally‎, ‎we compute the $n^{th}$ commutativity degree of a finite class‎ ‎of non-quasi-commutative semigroups and we conclude that it is less than $\dfrac{1}{2}$‎.
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