We provide a construction of monomial ideals in $R=K[x,y]$ such that $\mu \left(I^2\right)<\mu \left(I\right)$, where $\mu$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $\mu \left(I^k\right)$ that generalize some results...