Leavitt path algebras over a poset of fields

Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $le$ that satisfies $vle w$ whenever there exists a directed path from $w$ to $v$. Assuming that $I$ is a tree, we define a poset of fields over $I$ as a family $mathbf K = { K_i :iin I }$ of fields $K_i$ such that $K_isubseteq K_j$ if $jle i$. We define the concepts of a Leavitt path algebra $L_{mathbf K} (E)$ and a regular algebra $Q_{mathbf K}(E)$ over the poset of fields $mathbf K$, and we show that $Q_{mathbf K}(E)$ is a hereditary von Neumann regular ring, and that its monoid $mathcal V (Q_{mathbf K}(E))$ of isomorphism classes of finitely generated projective modules is canonically isomorphic to the graph monoid $M(E)$ of $E$.