Permutation functions and \((n,m)\)-commutativity of semigroups (Q1313477)

We say that a semigroup \(S\) is \((n,m)\)-commutative where \(n\), \(m\) are positive integers, if the identity \(x_ 1x_ 2 \dots x_ ny_ 1 y_ 2 \dots y_ m = y_ 1 y_ 2 \dots y_ m x_ 1 x_ 2 \dots x_ n\) is satisfied in \(S\). For a semigroup \(S\) define a function \(f_ S\) from the set \(N^ +\) of all positive integers into itself such that \(f_ S(n) = \min\{m;\;S \text{ is }(n,m)\text{-commutative}\}\) if there exists an \(m\) such that \(S\) is \((n,m)\)-commutative and \(f_ S(n)\) is not defined if such an \(m\) does not exist. For any semigroup \(S\) either \(f_ S\) is not defined for any \(n\), or \(f_ S(n)\) is defined for any \(n \in N^ +\). Let \(S\) be a semigroup such that \(f_ S\) is defined for any \(n \in N^ +\). Then the following four conditions hold: (i) \(f(n) = 1\) for all \(n > f(1)\); (ii) \(n + f(n) = f(1)\) or \(n + f(n) = f(1) + 1\) for all \(1 \leq n \leq f(1)\); (iii) if \(n + f(n) = m + f(m) = f(1)\) and \(m < f(n)\) for \(1 < n,m < f(1)\) then \(f(n + m) = f(n) - m\); (iv) if \(f(n) + n = f(1)\) then \(f(f(n)) = n\). On the other hand for any \(f\) satisfying these conditions there exists a semigroup \(S\) with \(f = f_ S\). The properties of functions satisfying (i)--(iv) are described.