Condition (K) for Boolean dynamical systems

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system (B, L, θ) with countable B and L satisfies Condition (K) if and only if every ideal of its C*-algebra is gauge-invariant, if and only if its C*-algebra has the (weak) ideal property, and if and only if its C*-algebra has topological dimension zero. As a corollary we prove that if the C*-algebra of a locally finite Boolean dynamical system with B and L countable either has real rank zero or is purely infinite, then (B, L, θ) satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the C*-algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable B and L.