On semisimple proto-Abelian categories associated to inverse monoids

Let G be a finite abelian group written multiplicatively, with $\hat{G}=G\bigsqcup \left\{0\right\}$ the pointed abelian group formed by adjoining an absorbing element 0. There is an associated finitary, proto-abelian category ${\mathrm{Vect}}_{\hat{G}}$, whose objects can be thought of as finite-dimensional vector spaces over $\hat{G}$. The class of $\hat{G}$-linear monoids are then defined in terms of this category. In this paper, we study the finitary, proto-abelian category $\mathrm{Rep}(M,\hat{G})$ of finite-dimensional $\hat{G}$-linear representations of a $\hat{G}$-linear monoid M. Although this category is only a slight modification of the usual category of M-modules, it exhibits significantly different behavior for interesting classes of monoids. Assuming that the regular principal factors of M are objects of $\mathrm{Rep}(M,\hat{G})$, we develop a version of the Clifford-Munn-Ponizovskiĭ Theorem and classify the M for which each non-zero object of $\mathrm{Rep}(M,\hat{G})$ is a direct sum of simple objects. When M is the endomorphism monoid of an object in ${\mathrm{Vect}}_{\hat{G}}$, we discuss alternate frameworks for studying its $\hat{G}$-linear representations and contrast the various approaches.