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 $v\le w$ 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 $\mathbf K = \{ K_i :i\in I \}$ of fields $K_i$ such that $K_i\subseteq K_j$ if $j\le i$. 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 $E$.

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