Let A be a semisimple modular annihilator Banach algebra and let LA be the left regular representation of A. We show how the strong radical of A is related to the strong radical of LA.

We compute the kernel of cup product of 1-dimensional cohomology classes for a group G acting trivially on Z or F2, by means of the naturality of cup product and the 5-term exact sequence of low degree of a suitable LHS spectral sequence. We determine thereby when cup product is injective, and when it is null.

The growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

The duality results of Wolfe for scalar convex programming problems and some of the more recent duality results for scalar nonconvex programming problems are extended to vector valued programs. Weak duality is established using a ‘Pareto’ type relation between the primal and dual objective functions.

In this paper we prove an analogue of the separable version of Nachbin's characterization of injective Banach spaces in the setting of Banach lattices. The mappings involved are continuous Riesz homomorphisms defined on ideals of separable Banach lattices which can be extended to Riesz homomorphisms on the whole Banach lattice. We discuss applications to simultaneous extension operators and to extension of continuous mappings between certain topological spaces.

An inverse semigroup S is said to be modular if its lattice 𝓛𝓕 (S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

A semigroup is eventually regular if each of its elements has some power that is regular. Let 𝓚 be one of Green's relations and let ρ be a congruence on an eventually regular semigroup S. It is shown for 𝓚 = 𝓛, 𝓡 and 𝓓 that if A and B are regular elements of S/ρ that are 𝓚-related in S/ρ then there exist elements a ∈ A, b ∈ B such that a and b are 𝓚-related in S. The result is not true for 𝓗 or 𝓙.

In the case when 0 < p < 1 it is proved, using a method of Macphail that the identity map i: lp → lp is not (r, s)-absolutely summing for any r, s.

For a doubly commuting n-tuple of unbounded normal operators in a Hilbert space the joint spectral measure can be constructed and its closed support described as the joint spectrum of the given n-tuple. The same is here shown for larger, possibly uncountable, families of operators.

We show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

A method for constructing Fitting-Schunck classes is given: the method is an adaptation of one given by C. L. Kanes for constructing Fitting formations, and generalizes the Fitting-Schunck class construction given by Cossey in 1981. A criterion for deciding which of the Fitting-Schunck classes so constructed are formations is given.

The class of all uniformly strongly prime rings is shown to be a special class of rings which generates a radical class which properly contains both the right and left strongly prime radicals and which is independent of the Jacobson and Brown-McCoy radicals.

The notion of (p, k)-epi mappings is introduced. The properties of such mappings are studied and the results obtained are applied to some differential equations.

K. D. Magill has investigated the semigroup generated by the idempotent continuous mappings of a topological space into itself and examined whether this semigroup determines the space to within homeomorphism. By analogy with this (and related work of Bridget Bos Baird) we now consider the semigroup generated by nilpotent continuous partial mappings of a space into itself.

Degeneracies occur with increasing frequency in some large scale linear programming problems, but with a simple change to the (revised) simplex method, resulting stationarity of the algorithm can be reduced. The method introduced here may also prevent cycling; neither the lexicographic refinement of Dantzig, Orden and Wolfe nor the perturbation technique of Charnes may be required to prevent cycling.