Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-gradedalgebra. We investigate the exponential rate of growth of the sequence ofgraded 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 ratecan be equal to arbitrarily large non-integers. An explicit formula is given.Surprisingly, this class consists of an infinite family of algebras simple asan $S$-graded algebra. This is in strong contrast to the group graded case forwhich the growth rate of such algebras always equals $\dim (A)$. In light ofthe previous, we also handle the problem of classification of all $S$-gradedsimple algebras, which is of independent interest. We achieve this goal for animportant class of semigroups that is crucial for a solution of the generalproblem.
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机译：令$ S $为有限半群，令$ A $为有限维$ S $ -gradedalgebra。我们研究了$ A $的梯度余维数$ c_n ^ S（A）$的序列的指数增长率，即$ \ lim \ limits_ {n \ rightarrow \ infty} \ sqrt [n] {c_n ^ S（A） } $。对于组评分，它始终是整数。最近在[20]中发现了第一个非整数增长的例子。我们提出了一大类代数，我们证明了它们的增长速度可以等于任意大的非整数。给出了一个明确的公式。令人惊讶的是，此类由无限个代数组成，这些代数是简单的$ S $阶代数。这与群体分级的情况形成了鲜明的对比，在这种情况下，此类代数的增长率始终等于$ \ dim（A）$。鉴于前面的内容，我们还处理了所有具有独立利益的$ S $阶简单代数的分类问题。我们针对重要的半群类实现了这一目标，这对于解决一般性问题至关重要。
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