Remarks on the sum of element orders of non-group semigroups

TThe invariant $\psi (G)$, the {\it sum of element orders} of a finite group $G$ will be generalized and defined for the finite non-group semigroups in this paper. We give an appropriate definition for the order of elements of a semigroup. As well as in the groups we denote the sum of element orders of a non-group semigroup $S$, which may possess the zero element and$/$ or the identity element, by $\psi (S)$. The non-group monogenic semigroup will be denoted by $C_{n,r}$ where $2\leq r\leq n$. In characterizing the semigroups $C_{n,r}$ we give a suitable upper bound and a lower bound for $\psi (C_{n,r})$, and then investigate the sum of element orders of the semi-direct product and the wreath product of two semigroups of this type. A natural question concerning this invariant may be posed as "For a finite non-group semigroup $S$ and the group $G$ with the same presentation as the semigroup, is $\psi (S)$ equal to $\psi (G)$ approximately?" We answer this question in part by giving classes of non-group semigroups, involving an odd prime $p$ and satisfying $\lim_{p\rightarrow \infty} \frac{\psi (S)}{\psi (G)}=1$. As a result of this study, we attain the sum of element orders of a wide class of cyclic groups, as well.