Σ -Functions of the Categories of Matrix Representations of Nilpotent Semigroups

We study the categories of matrix representations of nilpotent semigroups over an arbitrary field. Our main attention is given to the Auslander matrix algebras as one of possible forms of specifying the categories of representations, which have finitely many equivalence classes of indecomposable objects, and to the analysis of their discrete characteristics. The concept of Auslander algebra is generalized to the category of representations without auxiliary conditions. For any nilpotent cyclic semigroup, we describe its Auslander algebra and determine its Σ -function.