University of Guilan Journal of Algebra and Related Topics 2345-3931 10 2 2022 12 01 Remarks on the sum of element orders of non-group semigroups 113 129 6005 10.22124/jart.2022.22828.1424 EN Y. Marefat Department of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, Iran M. Gholami Department of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, Iran H. Doostie Department of Mathematics, University of Kharazmi, Tehran, Iran H. Refaghat Department of Mathematics, Islamic Azad University-Tabriz Branch, Tabriz, Iran Journal Article 2022 08 23 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. https://jart.guilan.ac.ir/article_6005_2d95dece65ace27a6ff1d46f093f3b7e.pdf