We show that if $T$ is any of four semigroups of two elements that are notgroups, there exists a finite dimensional associative $T$-graded algebra over afield of characteristic $0$ such that the codimensions of its graded polynomialidentities have a non-integer exponent of growth. In particular, we provide anexample of a finite dimensional graded-simple semigroup graded algebra over analgebraically closed field of characteristic $0$ with a non-integer gradedPI-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 thegraded analog of Amitsur's conjecture, i.e. there exists an integer gradedPI-exponent. Moreover, in the first case it turns out that the ordinary and thegraded PI-exponents coincide. In addition, we consider related problems on thestructure of semigroup graded algebras.
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机译：我们证明，如果$ T $是不是元素的两个元素的四个半群中的任何一个，则在特征$ 0 $的区域上存在有限维的关联的$ T $梯度代数，使得其梯度多项式恒等式的共维数为非整数增长的指数。特别是，我们提供了一个特征为$ 0 $的代数封闭域上具有有限整数的PI指数的有限维渐变简单半群渐变代数的示例，该指数严格小于代数的维数。但是，如果$ T $是左或右零波段，并且$ T $分级的代数是单位的，或者$ T $是可消半群，则$ T $分级的代数满足Amitsur猜想的分级类似物，即存在一个整数的分级PI指数。此外，在第一种情况下，事实证明普通PI指数和渐变PI指数是重合的。另外，我们考虑了半群级代数结构的相关问题。
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