Semilattices which must contain a copy of \(2^ N\) (Q1057889)

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 \textit{J. Liukkonen} and \textit{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.