Center of the semigroup algebra of a finite inverse semigroup over the field of complex numbers

In the semigroup algebra $A$ of a finite inverse semigroup $S$ over the field of complex numbers to an indempotent $e$ there is assigned the sum $\sigma(e)=e+\sum(-1)^ke_{i_1}\cdots e_{i_k}$, where $e_1,\dots,e_m$ are maximal preidempotents of the idempotent $e$, and the summation goes over all nonempty subsets $\{i_1,\dots,i_k\}$ of the set $\{1,\dots,m\}$. Then for any class $\mathscr K$ of conjugate group elements of the semigroup $S$ the element $K=\sum a\cdot\sigma(a^{-1}a)$ (the summation goes over all $a\in\mathscr K$) is a central element of the algebra $A$, and the set $\{K\}$ of all possible such elements is a basis for the center of the algebra $A$. Bibl. 2 titles.