Lattices of ideals in graph algebras: refining the join operation

We consider the ideal structure of Leavitt path algebras of graphs that satisfy Condition (K) over commutative rings with identity. The approach we use takes advantage of the relationship between Leavitt path algebras and Steinberg algebras of boundary path groupoids. We establish a lattice $\mathcal{F}'$ consisting of particular maps $\tau:E^0\to \LR$ and show that this lattice is isomorphic to the lattice of ideals of $L_R(E)$. The advantage to our approach over previous lattice isomorphisms, even in the case that $R$ is a field, is that we obtain convenient join and meet operations in $\mathcal{F}'$. Lastly, we provide three concrete examples.