Semigroup graded algebras and graded PI-exponent

Let S be a semigroup. We study the structure of graded-simple S-graded algebras A and the exponential rate PIexp S-gr(A):= lim n→∞cnS−gr(A)n of growth of codimensions c n S-gr (A) of their graded polynomial identities. This is of great interest since such algebras can have non-integer PIexp S-gr(A) despite being finite dimensional and associative. In addition, such algebras can have a non-trivial Jacobson radical J(A). All this is in strong contrast with the case when S is a group since in the group case J(A) is trivial, PIexp S-gr(A) is always integer and, if the base field is algebraically closed, then PIexp S-gr(A) equals dimA. Without any restrictions on the base field F, we classify graded-simple S-graded algebras A for a class of semigroups S which is complementary to the class of groups. We explicitly describe the structure of J(A) showing that J(A) is built up of pieces of a maximal S-graded semisimple subalgebra of A which turns out to be simple. When F is algebraically closed, we get an upper bound for lim¯n→∞cnS−gr(A)n. If A/J(A) ≈ M 2(F) and S is a right zero band, we show that this upper bound is sharp and PIexp S-gr(A) indeed exists. In particular, we present an infinite family of graded-simple algebras A with arbitrarily large non-integer PIexp S-gr(A).