One should apologize only to a person one has wronged, and only for having wronged that person  To apologize sincerely is to acknowledge that one has acted wrongfully  One cannot apologize sincerely unless one intends not to repeat that wrongful act
To accept an apology sincerely is to acknowledge a wrong, but also to vow not to hold a grudge against the wrongdoer
An apology cannot be both sincerely offered and sincerely accepted unless each person acknowledges that a wrongful act has occurred.