The purpose of this note is to prove that certain semilattices must contain a copy of \(2^ N\). We show that any locally compact non-Lawson semilattice must contain such a subsemilattice. This result has interesting ramifications in several areas. For example, it is well known that any locally compact semilattice of finite breadth is Lawson. Finite breadth is generalized by J. Liukkonen and M. Mislove [Lect. Notes Math. 998, 202-214 (1983; Zbl 0516.43001)] to the notion of compactly finite breadth, whereby the locally compact semilattice S has compactly finite breadth if every compact subset X of S has a finite subset \(F\subset X\) with inf F\(=\inf X\). Our result shows that any semilattice satisfying this condition must be Lawson. This completes the work in [loc. cit.] by showing that a locally compact semilattice S has compactly finite breadth if and only if every complex homomorphism of the measure algebra M(S) is given by integration against a universally measurable semicharacter of S.