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

In this paper, we solve a problem proposed by Melvyn Nathanson concerning the solutions to the the functional equations arising from multiplication of quantum integers, first studied by Nathanson, with field of coefficients Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} and whose supports are not prime sub-semigroups of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document}. As a consequence, we also obtain results concerning the problem of maximal solutions to these functional equations.