Divisible commutative semigroups

Abstract

Let N, R+ and R denote the set of all positive integers, the set of all positive real numbers and the set of all real numbers, respectively. Let (S,+) be a semigroup. If for any element x of S and for any positive integer n, there is an element y of S such that x = ny = y + ... + y (n times), then S is said to be divisible. A semigroup S is called power cancellative if and only if for x,y sigma S and n sigma N, nx = ny implies that x = y. In this research, we find necessary and sufficient conditions for subsemigroups of R+ under usual addition and R+ under usual multiplication to be divisible. We also prove a theorem on commutative power cancellative divisible semigroups. Moreover, we give examples of some noncommutative divisible subsemigroups of M2(R) under usual multiplication.