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Given an Artinian algebra $A$ over a field $k$, there are severalcombinatorial objects associated to $A$. They are the diagram $D_A$ as definedin [DK], the natural quiver $\Delta_A$ defined in \cite{Li} (cf. Section 2),and a generalized version of $k$-species $(A/r, r/r^2)$ with $r$ being theJacobson radical of $A$. When $A$ is splitting over the field $k$, the diagram$D_A$ and the well-known ext-quiver $\Gamma_A$ are the same. The main objectiveof this paper is to investigate the relations among these combinatorial objectsand in turn to use these relations to give a characterization of the algebra$A$.

Approach to artinian algebras via natural quivers