nLab directed join

Idea

A directed join is simply a join of a directed set.

Definitions

More precisely, if $P$ is a poset and $D$ is a subset of $P$, then we can consider the join $\bigvee D$ (if it exists) of $D$ in $P$. Since $D$ is a poset in its own right, we can also consider whether $D$ is directed set. If so, then $\bigvee D$ (if it exists) is a directed join in $P$. Sometimes one denotes that $\bigvee D$ is a directed join by making a little arrow out of the upper-right flank of the symbol (so it's a mix of ‘$\bigvee$’ and ‘$\nearrow$’). Unfortunately, I haven't found that symbol in LaTeX or Unicode. Possible workaround is $\bigvee{}^{\uparrow} D$.

A codirected meet in $P$ is a directed join in $P^\op$, but people don't talk about those so much.

By default, we mean finitely directed sets, that is $\aleph_0$-directed. If instead we take the join of a $\kappa$-directed set (for some regular cardinal $\kappa$), then we have a $\kappa$-directed join.

Properties

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 $\kappa$ elements and all $\kappa$-directed joins, then it is a suplattice.

A topological space (or locale) $X$ is compact if and only if $X$ may be expressed as a directed join of open subsets only trivially. That is, whenever $D$ is a directed collection of opens, if $X = \bigcup D$, then $X \in D$.

Directed colimits and filtered colimits are two slightly different categorifications of directed joins.

DCPOs

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.

A pointed dcpo is a DCPO with a bottom element (which is rather more specific than a pointed object in the category of DCPOs).

DCPOs are studied widely in domain theory.