It has been shown ([8], [2], [1], [3], [9]) that a collection of physical operations or experiments can be represented by a nonempty set of nonempty sets satisfying certain technical conditions. Such a set is called a manual. The operations in are looked at as having no “before” and no “after”, i.e., they are isolated in time. If we wish to look at connected sequences of operations—in particular, if we wish to condition by events in —we must look at the compound manual c whose elements represent compound operations built up from the operations in the base manual .

There is a large body of literature on inverse semigroups. This literature contains a considerable amount of information concerning congruences on these semigroups, which is not surprising in view of the demonstrated fact that congruences on inverse semigroups play a decisive role in most of the existing structure theorems. In addition, for an inverse semigroup of known structure, finding its congruence lattice, or even certain properties of this lattice, often gives information about these semigroups not apparent in their structure theorems.

Over the last few years, various extensions of the topological degree of a mapping have been made so as to include non-compact perturbations of the identity. One such extension, which employs compactness conditions, has been to the class of limit compact maps which were extensively studied by Sadovsky [7]. The class is a large one as it contains all compact mappings, contraction mappings and, more generally, condensing mappings. Sadovsky [7] gives a theory of degree for maps of the form I-f, where f is limit compact, and this was extended independently and with different methods by Petryshyn and Fitzpatrick [4] and the author [9] to allow f to be a multi-valued mapping. A refinement of the methods of [9] was given by Vanderbauwhede [8].

If R is a ring and S is a semigroup, the corresponding semigroup ring is denoted by R[S]. A ring is semiprime if it has no nonzero nilpotent ideals. A semigroup S is a semilattice P of semigroups Sα if there exists a homomorphism φ of S onto the semilattice P such that Sα = αφ−1 for each α ∈ P.

Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let Rbe a ring with 1. If His finite and G is not finitely generated we show that any non–zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.

Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t < p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a → ba, a → ab, a → ab + ba, a → bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms of which are weakly compact perturbations of the identity.

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p > 0, then kG is locally Noetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

The following theorem was proved in [1].

Theorem 1. Let S and T be continuous, commuting mappings of a complete, bounded metric space (X, d) into itself satisfying the inequality

for all x, y in X, where 0 ≤ c < 1 and p, p′, q, q′ ≥ 0 are fixed integers with p + p′, q + q′ ≥ 1. Then S and T have a unique common fixed point z. Further, if p′ or q′ = 0, then z is the unique fixed point of S and if p or q = 0, then z is the unique fixed point of T.

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.

Throughout this note all rings considered will be commutative and noetherian and will have non-zero identity elements. A will always denote such a ring and the category of all A-modules and all A-homomorphisms will be denoted by .

Let A be a complex Banach algebra with unit 1 satisfying ‖1‖ = 1. An element u in A is said to be unitary if it is invertible and ‖u‖ = ‖u−1‖ = 1. An element h in A is said to be hermitian if ‖exp(ifh)‖ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π (x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x) = y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

In [1] the following question was posed by R. W. Irving (see also Conjecture 4.10 in [4]): Is there an edge 2-colouring of the complete bipartite graph K13.17 with no monochromatic K3.3? We give a negative answer in this note (Theorem 2). Furthermore we prove Conjecture 4.11 (i) of [4] (Theorem 1), that is, any edge 2-coloured K2n+i,4n-3 contains a monochromatic K2,n with the 2 and n vertices a subset of the 2n +1 and 4n–3 vertices, respectively. Theorem 1 is a consequence of Satz 4 in [3], however, we give a direct proof here.

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

In part I of this paper P. Hall's formula for finite stem groups was derived. Using results of C. R. Leedham-Green and S. McKay, a similar formula for isoclinic groups with arbitrary branch factor group is shown.

The main result of this paper is the following theorem, which appears without proof in [1, p. 203].

Let fr:Xr → BO(r) be a sequence of fibrations with maps gr:Xr → Xr+1 such that the usual diagram commutes. For such a situation R. Lashof defines the concept of an X-structure on manifolds (see [3]), and proves a Thom-isomorphism for the cobordism groups of such manifolds. Let n, m be positive integers which are fixed throughout this paper. If r is very big in comparison with n + m then Xr is a simply connected CW-complex and the map (gr)*:H*(Xr; Q)→ H*(Xr+l; Q) is an isomorphism up to dimension n. Let γ be the pull-back over Xr of the universal r-linear bundle (which is, of course, a bundle over BO(r)). If r is very big in comparison with n + m, then we put X = Xr, and we assume that γ is orientable and oriented. The elements of H*(X; Q) of dimension not greater than n, will be called rational universal X-characteristic classes. It is well-known that many of the usual classes of manifolds may be described in terms of X-structures, (e.g. SO, SU, Spin-manifolds etc.).