when Al2O3 melts which bonds are broken? Previously, I was trying to understand why $\ce{Al2O3}$ has a lower melting point than MgO. An answer on quora said


  Do not confuse the character of the bond between the metal and oxygen with the bonds between molecules in the compound. The polar character of the bonds between aluminum and oxygen is indeed less than between magnesium and oxygen but this describes the energy needed to separate the metal from oxygen. When a solid melts it doesn’t break down the bonds between oxygen and aluminum or magnesium. The intermolecular bonds (that is bonds between one $\ce{Al2O3}$ and another $\ce{Al2O3}$) are in fact weaker than the bonds between two magnesium oxide molecules. This results in the greater melting point of magnesium oxide in spite of the less covalent character of the bonds within the molecule.


But if $\ce{Al2O3}$ is ionic and forms a giant lattice, which bonds that are "between one $\ce{Al2O3}$ and another $\ce{Al2O3}$" are broken?
Right, the text you are mentioning is not clear enough. It is not correct  to say that the molecule $\ce{Al_2O_3}$ exists and is separated from the next $\ce{Al_2O_3}$. Such a molecule does not exist. A sample of $\ce{Al_2O_3}$ is made of a superposition of $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ ions. It is not made of a superposition of molecules $\ce{Al_2O_3}$. Now the fact that MgO melts at a higher temperature than $\ce{Al_2O_3}$ is due to the stronger attraction between $\ce{Mg^{2+}}$ and $\ce{O^{2-}}$ ions. For some reason, the attraction between $\ce{Al^{3+}}$ ions and $\ce{O^{2-}}$ is weaker.