The infimum of a set is the greatest lower bound of and is denoted .

Let be a set with a partial order , and let . For any , is a lower bound of if for any . The infimum of , denothed , is the greatest such lower bound; that is, if is a lower bound of , then .

Note that it is not necessarily the case that . Suppose ; then , but .

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.