# Is there a name for this lattice theoretic property?

Suppose that $\langle L, \wedge, 0 \rangle$ is a lower semilattice with least element $0$. For preliminary notation, for all $b \in L$ define ${\downarrow}b = \{ a \in L \colon a \leq b \}$. Here $\leq$ is the partial order induced by the meet operation. The property I am interested in I call swathing. For all $s \in L$ and all $A \subseteq {\downarrow}s$ we will say that $s$ swathes $A$ if and only if: for all $b \in {\downarrow}s$ if, for all $a \in A$ we have $a \wedge b = 0$ then we also have $b = 0$.

My interest in this property stems from the following observation concerning convex sets in ${\mathbb{R}}^{n}$. The meet operation will be intersection. A convex set $O$ is open in the usual topology if and only if for all convex $C \subseteq {\mathbb{R}}^{n}$ if neither $C \cap O$ nor $C \smallsetminus O$ is empty then there exists a nonempty convex set $S \subseteq C \smallsetminus O$ that satisfies $O \cup S$ swathes $\{ O, S \}$.

Any use of the real numbers is hidden in the construction of the convex sets. Once you prove that if two convex sets $O_{0}$ and $O_{1}$ have this property then so does $O_{0} \cap O_{1}$. This property gives us a basis for the usual topology.

With a little additional fussiness we can work in a semilattice as described above and construct what amounts to a basis for a locale.

Is there an commonly used name for this property.

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Nothing in mathematics is official. Terms are commonly used, or accepted, or standard. But there is no higher authority telling us what to call things. – Willie Wong Nov 30 '11 at 10:31
In On the duality of cardinal invariants, Top. Proc. 2, no. 2 (1977) pp. 563-582 I used the ad hoc terminology $A F\text{-}covers\; s$, but that was chosen to suit a very specific context. Outside that context my inclination would be to say that $A$ is dense in (or below) $s$. – Brian M. Scott Nov 30 '11 at 16:35
Bowing to Willie Wong's authority ;), I have made the change from official to commonly used. – Jay Dec 1 '11 at 1:05
Brian M. Scott, it appears there is no commonly used name. Your suggestions of dense in or dense below seem to work for my purposes. If you want to, change your comment into an answer. – Jay Dec 1 '11 at 1:14

In On the duality of cardinal invariants, Top. Proc. 2, no. 2 (1977) pp. 563-582 I used the ad hoc terminology $A F\text{-}covers\;s$, but that was chosen to suit a very specific context. Outside that context my inclination would be to say that $A$ is dense in (or below) $s$. I don’t know any standard term, but I’ve not worked much in this area, and there could certainly be one of which I’m not aware.