Translational hull and semigroups of binary relations
Mario Petrich · Glasgow Mathematical Journal · 1968
Various semigroups of partial transformations (and more generally, semigroups of binary relations) on a set have been studied by a number of Soviet mathematicians; to mention only a few: Gluskin [2], Ljapin [4], Shutov [6], Zaretski [7], [8]. In their study the densely embedded ideal of a semigroup introduced by Ljapin [4] plays a central role. In fact, a concrete semigrouQis described in several instances by its abstract characteristic, namely either by a set of postulates on an abstract semigroup or by a set of postulates (which are usually much simpler) on an abstract semigroup S which is a densely embedded ideal of a semigroupTisomorphic toQ. In many cases, the densely embedded idealSis a completely 0-simple semigroup. The following theorem [3, 1.7.1] reduces the study of a semigroupQwith a weakly reductive densely embedded ideal S to the study of the translational hull ofS: Theorem (Gluskin).If S is a weakly reductive densely embedded ideal of a semigroup Q, then Q is isomorphic to the translational hullω(S)of S.