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

Tthe invariant , the {it sum of element orders} of a finite group 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 , which may possess the zero element and or the identity element, by . the non-group monogenic semigroup will be denoted by where . in characterizing the semigroups we give a suitable upper bound and a lower bound for , 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 and the group with the same presentation as the semigroup, is equal to approximately? we answer this question in part by giving classes of non-group semigroups, involving an odd prime and satisfying . as a result of this study, we attain the sum of element orders of a wide class of cyclic groups, as well.