If your marriage ended in divorce, the romantic aspect is over, however the relationship can continue if you financially support your ex via alimony.
Traditionally, alimony payments cease if and when the former spouse remarries. However, in an attempt to maintain alimony for as long as possible, ex-spouses often steer clear of marriage. This situation of having a long-term live-in boyfriend or girlfriend, otherwise known as cohabitation, can be grounds for reducing or eliminating alimony.