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Modulation and natural valued quiver of an algebra

The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural valued quiver}, of an artinian algebra so as to correspond to its valued Ext-quiver when this algebra is not $k$-splitting over the field $k$. Moreover, we illustrate the relation between the valued Ext-quiver and the natural valued quiver. The interesting fact we find is that the representation categories of a pseudo-modulation and of a pre-modulation are equivalent respectively to that of a tensor algebra of $\mathcal A$-path type and of a generalized path algebra. Their examples are given respectively from two kinds of artinian hereditary algebras. Furthermore, the isomorphism theorem is given for normal generalized path algebras with finite (acyclic) quivers and normal pre-modulations. Four examples of pseudo-modulations are given: (i) group species in mutation theory as a semi-normal generalized modulation; (ii) viewing a path algebra with loops as a pre-modulation with valued quiver which has not loops; (iii) differential pseudo-modulation and its relation with differential tensor algebras; (iv) a pseudo-modulation is considered as a free graded category.