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→∞ \(\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\) 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
\({\overline {\lim } _{n \to \infty }}\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\). 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).
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