Nathanson quantum functional equations and the non-prime semigroup support solutions

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

“…• Problem 4 in [4] when the fields of coefficients are of characteristic zero and the supports A are semi-cyclic but not necessarily prime subsemigroups of N. Together with [9], these results provide a complete solution to Problem 1 for the case of rational fields of coefficients and supports A, subsemigroups of N not necessarily prime subsemigroups. • Problem 3 in [4] concerning maximal solutions (see also [5,8]) when the fields of coefficients are of characteristic zero and the support A and A are both semi-cyclic subgroups of N but not necessarily prime subsemigroups.…”

“…In this paper, we solve an open problem proposed by Nathanson and Wang concerning the classification of solutions with supports which are not prime subsemigroups of N. Together with [9], our results in this paper also provide a complete solution to this problem for the case of rational field of coefficients. An overview of the topic and some relevant background concerning quantum integers and the functional equations arising from their multiplication is given in this section.…”

“…Many results concerning solutions whose semigroup supports are prime subsemigroups of N are provided in [4]- [12], and it can be seen from these papers that obtaining such results requires the existences of prime-indexed elements in these solutions. Because of the lack of prime-indexed elements in the solutions whose semigroup supports are not prime subsemigroups of N, few results are known [9] concerning these solutions. For the case considered in this paper, we have to employ a completely different method from those used in [4]- [12] to obtain our results.…”

“…The results of this paper and those of [9] are also used in [12] to resolve another problem in this area concerning extensions of supports, not necessarily prime subsemigroups of N, of polynomial solutions with fields of coefficients K of characteristic zero. Moreover, we plan to establish in a future paper the analogues of the results in [9] for the rational function solutions with non-prime semigroup supports and fields of coefficients of characteristic zero (see [10,11] for the existence and characterization of rational function solutions with prime semigroup supports and fields of coefficients of characteristic zero).…”