On page 18 of this paper, the author states that there is a duality (correspondence?) between semilattices (i.e., abelian semigroups of idempotents) and totally disconnected locally compact Hausdorff spaces.
Given a semilattice $E$, consider the space of characters
$$\widehat{E} := \{\chi : E \to \{0,1\} \mid \chi \neq 0 \text{ semigroup homomorphism} \}.$$
equipped with the topology of pointwise convergence. From what I gather, the topology of pointwise convergence on $\widehat{E}$ is the topology it inherits in virtue of being a subspace of the topological space $\{0,1\}^{E}$. Since $\{0,1\}^{E}$ is totally disconnected and Hausdorff (a product of totally disconnected spaces is totally disconnected; same is true for Hausdorffness), $\widehat{E}$ is also totally disconnected and Hausdorff. Moreover, $\{0,1\}^E$ is compact and therefore locally compact (compact Hausdorff spaces are locally compact).
However, I am having trouble showing that $\widehat{E}$ is locally compact. Also, what about the other direction. If one starts with a totally disconnected locally compact Hausdorff space $X$, how does one construct a semilattice?