When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition, it is well known that groups can be defined in terms of those two divisions. The aim of this article is to extend those results to other classes of unary semigroups. In the first part of the article, we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part, we solve a problem that was posed elsewhere. The article closes with a list of open problems.

CEMAT - Center for Computational and Stochastic Mathematics