Nest Representations of TAF Algebras

A nest representation of a strongly maximal TAF algebra $A$ is a representation $\pi$ for which $\operatorname{Lat} \pi(A) is totally ordered. We prove that if the spectrum of $A$ is totally ordered, or if $\mathrm{Lat}\pi (A)$ contains an atom, then $\ker \pi$ is a meet irreducible ideal.