University of GuilanJournal of Algebra and Related Topics2345-393110220221201Remarks on the sum of element orders of non-group semigroups113129600510.22124/jart.2022.22828.1424ENY.MarefatDepartment of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, Iran0000-0003-1156-6707M.GholamiDepartment of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, IranH.DoostieDepartment of Mathematics, University of Kharazmi, Tehran, IranH.RefaghatDepartment of Mathematics, Islamic Azad University-Tabriz Branch, Tabriz, IranJournal Article20220823TThe 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