The Probability that Two Semigroup Elements Commute Can Be Anything (original) (raw)

2015

In a recent article [2] in this Journal, Givens defined a finite semigroup’s commuting probability as the probability that x? y = y? x when x and y are chosen at random (independently and uniformly) from the semigroup elements. She asked which commuting probabilities can be achieved, and partially answered this question by showing that the achievable commuting probabilites are dense in (0, 1]. We extend this result to prove that every rational number in (0, 1] can be achieved. We begin by recalling Lagrange’s celebrated four-square theorem (found in, e.g., [1]). It states that every natural number can be expressed as the sum of four integer squares; furthermore, three squares suffice unless the number is of the form 4k(8m+ 7). The proof proceeds with four constructions; it is unknown if a single semigroup family can answer this question. Claim 1 Every rational in (0, 1/3] is an achievable commuting probability. Proof. For positive integers a, b, c and nonnegative integer k, we consi...