What is the ruling on breaking a supererogatory fast to accept a food invitation, and what if the day of the invitation coincides with a habitually observed fasting day, such as Monday and Thursday?

It is permissible for a Muslim to break a voluntary fast if he intends to do so, and it is recommended for him to complete it unless there is an excuse or a legitimate interest that necessitates breaking it. The fasting person is rewarded for his intention to break the fast for a good reason, though not as much as one who actually fasted.

As for the intention to break the fast to accept an invitation tomorrow, this prevents the formation of the intention to fast on that day. However, fasting does not prevent one from accepting an invitation; it is permissible for the fasting person to supplicate for the host without breaking his fast, unless he knows that the host has gone to great lengths for his sake, in which case it is recommended for him to break his voluntary fast and eat with him.