When one party is awarded permanent alimony, generally speaking those alimony payments will continue until death of either party or remarriage. Some people put language in their marital settlement agreements that states that alimony will conclude when the recipient of the alimony cohabits with a member of the opposite sex. The question then becomes what is the definition of "cohabitation". In Florida, I believe that most of the courts have stated that cohabitation would have to include a sharing of finances. The New York courts are dealing with this issue, and I'm sure its just a matter of time before all courts in all states set forth a clear definition of "cohabitation" so that people are not litigating this issue in the future.
To read about New York's take on this issue, see: http://www.newsday.com/news/local/wire/newyork/ny-bc-ny--definingcohabitat1021oct21,0,5977746.story