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 ۲\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)}=۱. As a result of this study, we attain the sum of element orders of a wide class of cyclic groups, as well.

Keywords:

Sum of element orders , Finite group , Non-group semigroups