Identities for a Class of Regular Unary Semigroups

Abstract

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° an inverse subsemigroup of S. Then S° is called an “inverse transversal of S” if it contains a unique inverse x° 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° as a regular unary semigroup (S, ○) with a regular unary operation “○” on S firstly. Then we prove that S is a regular semigroup with an inverse transversal S° if and only if (S, ○) satisfies the following identities (IST):

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 “○” 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.

Key Words:

• e-variety
• Inverse transversal
• ist-operation
• ist-semigroup
• ist-variety

2000 Mathematics Subject Classification:

• 20M

ACKNOWLEDGMENTS

This project has been supported by the National Natural Science Foundation of China and by the Provincial Natural Science Foundation of Guangdong Province.

Notes

Communicated by P. Higgins.