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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.
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