The translational hull Ω(S) of a semigroup S plays an important role in the theory of ideal extensions of semigroups. In fact, every ideal extension of S by a semigroup T with zero can be constructed using a certain partial homomorphism of T\0 into Ω(S); a particular case of interest is when S is weakly reductive (see §4.4 of [3], [2], [7]). A theorem of Gluskin [6, 1.7.1] states that if S is a weakly reductive semigroup and a densely embedded ideal of a semigroup Q, then Q and Ω(S) are isomorphic. A number of papers of Soviet mathematicians deal with the abstract characteristic (abstract semigroup, satisfying certain conditions, isomorphic to the given semigroup) of various classes of (partial) transformation semigroups in terms of densely embedded ideals (see, e.g., [4]). In many of the cases studied, the densely embedded ideal in question is a completely 0-simple semigroup, so that Gluskin's theorem mentioned above applies. This enhances the importance of the translational hull of a weakly reductive, and in particular of a completely 0-simple semigroup. Gluskin [5] applied the theory of densely embedded ideals (which are completely 0-simple semigroups) also to semigroups and rings of endomorphisms of a linear manifold and to certain classes of abstract rings.

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 semigrou Q is 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 semigroup T isomorphic to Q. In many cases, the densely embedded ideal S is a completely 0-simple semigroup. The following theorem [3, 1.7.1] reduces the study of a semigroup Q with a weakly reductive densely embedded ideal S to the study of the translational hull of S:

Theorem (Gluskin). If S is a weakly reductive densely embedded ideal of a semigroup Q, then Q is isomorphic to the translational hull ω(S) of S.

The lower radical of a module type. For a ring R with unit, the module type t(R) was defined in [6] as follows: t(0) = 0; t(R) = d if every free R-module has invariant rank; t(R) = (c, k) for integers c, k ≧ 1 if every free R-module of rank < c has invariant rank, while a free module of rank h ≧ c has rank h + nk for any integer n ≧ 0. The module types form a lattice under the ordering 0 < (c, k) < d and (c', k') ≦ (c, k) if and only if c ' ≦ c and k' Ⅰ k. Two of the basic theorems on types are:

A. [6; Theorem 2, p. 115] If R → R' is a unit-preserving homomorphism, then t(R')≦ t(R).

It is the purpose of this note to discuss the solution of the pair of series

where F(x) and G(x) are given and the coefficients an are to be determined.

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chain

then S is called an ω-semigroup.

In recent years there have appeared solutions of several integral equations of the type

in which the kernel K(x) contains (as a factor) one of the classical orthogonal polynomials or a hypergeometric function.