The $n^{th}$ commutativity degree of semigroups
Subject Areas : 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)
Keywords: commutativity degree, Probability, Quasi-commutative semigroups,
Abstract :
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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