Two Semigroup Elements Can Commute with Any Positive Rational Probability

Limit points in the range of the commuting probability function on finite groups

“…This is in striking contrast to the situation for semigroups, see [16]. In the case of groups, further insight into the range of Pr…”

On commuting probabilities in finite groups and rings

“…Givens [14] showed that the commuting spectrum for semigroups is dense in the interval [0, 1]. Later Ponomarenko and Selinski [26] proved that for any rational number in r ∈ (0, 1], there is a finite semigroup S such that the commuting probability in S is equal to r. Soule [29] found a single family of semigroups that has this property. These semigroups are defined as follows.…”

“…In 1995 Lescot [21], rederived classification of groups with commuting probability > 1 2 , using the notion of isoclinism in groups introduced by Hall [17]. Recently, the commuting probability in semigroups has been studied in [14], [24], [26] and [29].…”

On commuting probabilities in finite groups and rings

“…There has also been interest in the study of commuting probability of other algebraic structures; MacHale [23], investigated the notion of commuting probability in rings. Commuting probability in semigroups has been studied in [14], [24], [26] and [29].…”

“…Givens [14] showed that the commuting spectrum for semigroups is dense in the interval [0, 1]. Later Ponomarenko and Selinski [26] proved that for any rational number in r ∈ (0, 1], there is a finite semigroup S such that the commuting probability in S is equal to r. Soule [29] found a single family of semigroups that has this property.…”