A note on inverse semigroups

The concept of an 'inverse semigroup' or 'generalized group' has recently been introduced, independently, by Preston (4) and Vagner (5). In the present note we give several alternative definitions for such a system, some of which appear to be weaker than that originally given. A semigroup is a non-empty set *S which is closed with respect to an associative binary operation. An element e of 8 is termed an idempotent if e = e. A left ideal LofS is a non-empty subset of 8 such that SL £ L, and a right ideal R of S is a non-empty subset of 8 such that RS c R. A two-sided ideal (or, simply, an ideal) of S is a subset which is both a left and a right ideal. Following Green (3), we shall say that the elements a, b of 8 are l-equivalent or t-equivalent according as a and b generate the same left or right ideal* of 8 respectively, and we write a = b(l) or a = b(v). If a left or right ideal is generated by a single element, it is termed principal. A semigroup S is said to be regular if for any given element a e S there is at least one solution of the equation axa = a in S. We observe that in a regular semigroup the equations axa = a> xax = x