It is well-known that, in any semigroup S, all the left and right Schützenberger groups of the ℋ-class contained in a fixed D class D of S are isomorphic to one group. We prove a sharper result: that, up to actionpreserving isomorphisms, all these Schützenberger groups and their classical isomorphisms are but one group and its identity isomorphism; thus, one group is essentially sufficient to describe not only all the Schützenberger groups of ℋ-classes in D, but their action on these ℋ-classes and classical isomorphisms as well.

Various investigations on (E, M)-categories are given by many authors. Recently, Herrlich (to appear) and Strecker (1972) gave interesting results on relations between factorizations (E, M) on nice categories and epireflective subcategories of . Factorization (E, M) treated by them satisfy that E ⊂ Epi. In this paper we shall consider factorization structures (E, M) such that E⊃Epi in the category of all topological spaces and study relations between these structures and some subcategories of . Our results are quite different from those in Herrlich (to appear) and Strecker (1972) and are rather related with the results in Herrlich (1969). Herrlich (1969) defined limit operators on and gave a one-to-one correspondence between suitable limit operators and bicoreflective subcategories of . We shall consider two kinds of factorization structures on and show that the first ones correspond to bicoreflective subcategories which are closed under closed embeddings, the second ones correspond to bireflective subcategories which satisfy that the classes of all reflections are closed under closed embeddings (see Proposition 11) and these correspondences are one-to-one.

The biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

Free groups and free inverse semigroups are characterized by certain properties which are usually called universal. Category-theoretically they arise from left adjoints of forgetful functors which assign to each structure its underlying set. The purpose of this note is to give a construction of a wealth of inverse semigroups with certain universal properties which subsumes free groups and free inverse semigroups and incidentally elucidates some well-known constructions of free inverse semigroups (Scheiblich (1972)). Categorytheoretical terminology will be freely taken from (Mac Lane (1971)).

Every lattice generated by three unordered elements contains a finite sublattice generated by three unordered elements. A list ℒ of twelve finite lattices, each generated by a three-element unordered set, is given. It is proved that every lattice generated by a three-element unordered set contains a sublattice isomorphic to onė of the lattices in ℒ moreover, ℒ is the smallest such list.

Let R be a normal subgroup of the free group F, and set G = F/[R, R]. We assume that F/R is a torsion-free group which is either solvable and not cyclic, or has a non-trivial center and is not cyclic-by-periodic. Then any automorphism of G whose restriction to R/[R, R] is trivial is an inner automorphism, determined by some element of R/[R, R]. This result extends a theorem of Šmel'kin (1967).

Starters with adders, in abelian groups of odd order, have been used widely in recent combinatorial constructions, most notably in the study of Room squares. In this paper, constructions of adders for the so-called “patterned starter” are described, for a large class of nonabelian groups of odd order.

The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra 〈A, f0, f1, f2, …〉 where f0, f1, f2, … is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operations F0, F1, F2, … over 〈B,·〉 such that 〈A, f0, f1, …〉 is a subalgebra of 〈B, F0, F1, …〉 satisfying exactly the same identities. Moreover, if there are only finitely many f0, f1, …, then 〈B1〉 can be taken so that its identities are recursive in those of 〈A, f0, f1, …〉, If 〈A, f0, f1, …〉 is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.

Generalization to arbitrary join-extensions of a poset of a recent characterization of those which are complete lattices.

Let ℒ be a normal base of a Tychonoff space X and ωℒ denote the Wallman-type (real-) compactification of X generated by ℒ. This Wallman-type compactification is known to associate with a unique proximity δ. A ℒ-filter ℒ is round if for each F ∈ ℒ there is an Fo ∈ ℒ there is an Fo ∈ ℒ such that Fo(X-F). A subset A of ω(£) is called a round subset of ω (£) iff for each Z ∈ ℒ, if C1w(x)Z contains A, then it is a neighborhood of A. Properties of round ℒ-filters and round sets of ω(ℒ) are introduced. We also prove that the intersection of all the free ℒ-ultrafilters is ℒ= {Z ∈ ℒ: C1x(X-Z) is compact} iff ω(ℒ) – X is a round subset of ω(ℒ) if ℒ is a separating nest generated intersection ring with property (α) then ω(ℒ) - v(ℒ) is a round subset of ω(ℒ).

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

Duchon (1967, 1969) and Duchon and Kluvánek (1967) considered the problems of the existence of countably additive tensor products for vector measures. Duchon and Kluvánek (1967) showed that a countably additive product with respect to the inductive tensor topology always exists while Kluvánek (1970) presented an example which showed that this was not the case for the projective tensor topology. Kluvánek considered yet another tensor product topology which is stronger than the inductive topology but weaker than the projective tensor topology and showed that a countably additive product for two vector measures always exists for this particular tensor topology, see Kluvánek (1973). He has conjectured that this topology is the strongest tensor topology (given by a cross norm) which always admits products for any two arbitrary vector measures. In this note we use an example of Kluvánek (1974) to show that this conjecture is indeed true when one of the factors in the tensor product is l2 and the other factor is metrizable. The construction used also clarifies a conjecture made by Swartz (to appear) concerning products of Hilbert space valued measures.

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with