Identities for a class of regular unary semigroups

A class V of regular semigroups is an e-variety if it is closed under homomorphic images, regular subsemigroups, and direct products. Let S be a regular semigroup and S degrees an inverse subsemigroup of S. Then S degrees is called an "inverse transversal of S" if it contains a unique inverse x degrees of each element x of S. Many important classes of regular semigroups form e-varieties of regular semigroups. However, the class of regular semigroups with inverse transversals does not form an e-variety. In this article, we consider a regular semigroup S with an inverse transversal S degrees as a regular unary semigroup (S, circle) with a regular unary operation "circle" on S firstly. Then we prove that S is a regular semigroup with an inverse transversal S degrees if and only if (S, circle) satisfies the following identities (IST): (IST) aa degrees a = a, a degrees aa degrees = a degrees, (a degrees b degrees)degrees = b degrees degrees a degrees degrees, aa degrees bb degrees aa degrees = aa degrees bb degrees, a degrees ab degrees ba degrees a = b degrees ba degrees a. Such a regular operation is called an "ist-operation," and a regular semigroup S is called an "ist-semigroup" if there exists an ist-operation "circle" on S. A regular subsemigroup T of a regular semigroup S is called an "ist-subsemigroup" if T is an ist-semigroup. A class V of ist-semigroups is an ist-variety if it is closed under homomorphic images, ist-subsemigroups, and direct products. We characterize the set of identities of (IST) and investigate the relationship among those identities. Also, we describe the classes of regular unary semigroups which satisfy some of these identities in (IST). On the basis of that, we'll characterize the ist-varieties, in a later article.