More precisely, if is a poset and is a subset of , then we can consider the join (if it exists) of in . Since is a poset in its own right, we can also consider whether is directed set. If so, then (if it exists) is a directed join in . Sometimes one denotes that is a directed join by making a little arrow out of the upper-right flank of the symbol (so it's a mix of ‘’ and ‘’). Unfortunately, I haven't found that symbol in LaTeX or Unicode. Possible workaround is .
A codirected meet in is a directed join in , but people don't talk about those so much.
By default, we mean finitely directed sets, that is -directed. If instead we take the join of a -directed set (for some regular cardinal ), then we have a -directed join.
If a join-semilattice (a poset with all finitary joins) has all directed joins, then it has all joins (and so is a suplattice, equivalently a complete lattice). More generally, if a poset has all joins of fewer than elements and all -directed joins, then it is a suplattice.
A poset which has all directed joins is called a directed-complete partial order, or dcpo. The homomorphisms of DCPOs are those functions that preserve directed joins; these are also called Scott-continuous because they are precisely the continuous maps relative to the Scott topology on the DCPOs.
DCPOs are studied widely in domain theory.