If $(X,R)$ is an ordered set then we say $R$ is an order in $X$, how do we call the relation $R$ if $(X,R)$ is a directed set? A direction in X?
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What do you mean by 'directed' here? It is a special partial order, isn't it? Then I would refer to $R$ as the partial order of the directed set. – Berci Jun 14 '16 at 13:50
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@Berci No, it is not a special partial order since in a direct set the antisymmetry does not need to hold, so it is not true that every directed set is a poset. A set is directed if it is equipped with a preorder and the aditional property that every pair of elements of the set is bounded above. – la flaca Jun 14 '16 at 14:54