The translational hull of an inverse semigroup
Glasgow Mathematical Journal
An ideal extension of one semigroup by another is determined by a partial homomorphism into the translational hull of the first semigroup [3, §2, Theorem 5]. In most instances, the development of the theory of ideal extensions has been hindered by inadequate knowledge of the translational hull; it is our purpose here to characterize certain basic structures in the translational hull of an arbitrary inverse semigroup. For an inverse semigroup 5, the translational hull of 5, £2 (5) , is again an
... (5) , is again an inverse semigroup, and thus the idempotents of £2(5) form a semilattice. How the structure of this semilattice, E a ( S) , is influenced by the structure of the semilattice of idempotents of 5, E s , is seen in one of our main results: E a(S ) cs &(E S ). Since £2(5) always possesses an identity, the group of units of £2 (5) is another structure which is of interest. We give here a characterization of this group in terms of automorphisms of the semilattice of 5. There are two sections dealing with applications of the characterizations given for E n(S) and the group of units of £2(5).