Completely simple semigroups form a variety, , of algebras with the operations of multiplication and inversion. It is known that the mapping , where is the variety of all groups, is an isomorphism of the lattice of all subvarieties of onto a subdirect product of the lattice of subvarieties of and the interval . We consider embeddings of into certain direct products on the above pattern with rectangular bands, rectangular groups and central completely simple semigroups in place of groups.

We show that stochastic compactness of partial sums with no normal limit distribution corresponds to stochastic compactness of the point processes generated by the observations so that there exist joint limit distributions for the sample sums and the sample maxima.

Let Γ be a graph with isomorphic subgraphs G and H, and let θ: G → H be an isomorphism. If θ can be extended to an automorphism of Γ, we call θ a partial automorphism of Γ.

We consider the application of partial automorphisms to the graph reconstruction conjecture, in particular, to the problem of reconstructing graphs with two vertices of degree k – 1 and the remaining vertices of degree k.

A verbal product is introduced for a particular class of varieties of inverse semigroups and this product is shown to be associative. As well, the structure of this class is examined.

We discuss the projective geometry defined in terms of the hollow factor modules of a given module. In particular, we derive an explicit expression for the division ring obtained in coordinatizing such a projective geometry.

Several procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

Connexions are sought between the subvarieties of a variety U of groups and the subvarieties of the variety of all groups which are central extensions by groups in U, in the case when U has the form . Here , is the variety of abelian groups of exponent dividing r and Bis a variety of soluble groups of finite exponent.

We prove that for a non-discrete space X, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i) X is paracompact and each point is Gδ, or (ii) X is a completely paracompact Morita k-space. Consequently, if X is a non-discrete completely paracompact space in which each point is a Gδ-set or it is also a Morita k-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a space X there exists a point x ∈ X and a family ϕ of supports on X such that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result for Rn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.

V. Krishnamurthy has shown that on a finite set X all topologies can be mapped into a certain set of matrices of zeros and ones. In this paper it is shown that all lattices, algebras and rings on a finite set X can be mapped onto particular sets of matrices of zeros and ones.

An elementary proof is given of a theorem of Castro and Lazer that the degree of the gradient of a coercive functional on a large ball of Rn is one.

The topology of the Čech fundamental group of the one-point compactification of an appropriate space Y induces a topology on the fundamental group of Y. We describe this topology in terms of a topological group introduced by Higman.