Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T-graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T-graded algebra is unital, or T is a cancellative semigroup, then the T-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.