Semigroup graded algebras and graded PI-exponent

Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an $S$-graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals $\dim (A)$. In light of the previous, we also handle the problem of classification of all $S$-graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of the general problem.