A join of the empty family (where $ I = \emptyset $ ) is an initial object of the category $ A $ , as in Example X. Equivalently, it is a least element of $ A $ : an element $ 0 \in A $ such that $ 0 \leq a $ for all $ a \in A $ . For instance, in $ (\reals, \mathord{\leq}) $ , join is supremum and there is no least element.