Any preorder P on a set X has an associated preorder P′, P″, P‴, … The proerties of this sequence are studied. When X is finite the sequence is eventually periodic with period P = 1 or p = 1, the eventual constant preorder is full p = 2 the possible forms which the eventual alternating order can take are examined: first, the possible combinations of components are enumerated; second, the notion of ramification at a caste is used to show that X may in a heuristic sense be of unbounded complexity. If X is orderdense the periodicity starts at P′.

A unified theory of continuous and certain non-continuous functions is proposed and developed. The proposed theory encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, almost continuous functions, c-continuous functions, c*-continuous functions, s-continuous functions, l-continuous functions, H-continuous functions, and the ε-continuous functions of Klee.

Starting with a class ℳ of Ω-groups, necessary and sufficient conditions on ℳ are given to ensure that the corresponding Hoehnke radical ρ (determined by the subdirect closure of ℳ as semisimple class) is a radical in the sense of Kurosh and Amitsur; has a hereditary semisimple class; satisfies the ADS-property; has a hereditary radical class or satisfies ρN ∩ I ⊆ ρI and lastly, have both a hereditary radical and semisimple class or satisfies ρN ∩ I = ρI.

Finite ρ-groups with all of their maximal subgroups isomorphic are studied by means of the coclass. All such groups of coclass I and 2 are determined, while those of coclass 3 are shown to have order at most ρ13. A general bound for the order is given as a function of ρ and the coclass only.

In the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

In this paper we consider classes of vector lattices over subfields of the real numbers. Among other properties we relate the archimedean condition of such a vector lattice to the uniqueness of scalar multiplication and the linearity of l-automorphisms. If a vector lattice in the classes considered admits an essential subgroup that is not a minimal prime, then it also admits a non-linear l-automorphism and more than one scalar multiplication. It is also shown that each l-group contains a largest archimedean convex l-subgroup which admits a unique scalar multiplication.

Let {λj}j≥0 be a sequence of positive integers such that λj+1/λj≥3 and {aj}j≥0 a sequence of complex numbers such that |aj|≤1. Let μ be the Riesz product πj≥0[1+ Re(ajeiλjx)], that is, the weak limit of measures on T the density of which are the partial products. Then if Σj≥0|aj|2≤∞ the series Σj≥0 aj(eiλjx - ½āj) converges for μ-almost every x. The μ-a.e. convergence of series Σ ajeinλjx is also investigated as well as the case of Riesz products on a compact commutative group.

W. Rudin has proved that the union of the Riesz set N ⊆ R with a Λ(l)-subset of Z is again a Riesz set. In this note we generalize his result to compact groups whose contains a circle group, thereby extending an earlier F. and M. Riesz theorem for such groups by the author. We also investigate the possibility of constructing Λ(p)-sets for these groups, departing from Λ(p)-sets for the circle group in center.

A collection P of bounded linear operators in l2 is constructed in such a manner that given any separable metric space X, and any countable collection F of continuous self-maps of X, there is a homeomorphism h of X onto a subset of l2 such that for each f ∈ F there is P ∈ P with hf = Ph.

While similar results were obtained by Baayen and Dc Groot, our construction makes it possible to impose additional conditions on h (depending on F). For example, if all the members of F are uniformly continuous then h too can be made uniformly continuous.

A semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

Examples are given that show the following: (1) normal structure need not be inherited by quotient spaces; (2) uniform normal structure is not a self-dual property; and (3) no degree of k–uniform rotundity need be present in a space with uniform normal structure.

Let G be a finite group of even order coprime to 3. If G admits a fixed-point-free automorphism group isomorphic to the symmetric group on three letters, then we prove that G is soluble.

In 1985 John Reade determined the spectrum of C1 regarded as an operator on the space c0 of all null sequences normed by ║x║ = supn≧0|xn|. It is the purpose of this paper to determine the spectrum of C1 regarded as an operator on the space bv0 of all sequences x such that xk → 0 as k → ∞ and .

Focussing on complete regularity, we discuss the separation properties of bitopological spaces. The unifying concept is that of separation by a pair of bases (B1, B2) for the closed sets of a bitopological space (S, J1, J2). For various separation properties a characterization is presented in terms of separation by a pair of closed bases. This is extended to results concerning pairs of subbases. Here the notion of screening by pairs of subbases plays a central role and the characterization of complete regularity in a natural way fits in between those of regularity and normality. In the key lemma the relation with quasi-proximities is exhibited.

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

Consider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

The n–th member of the growth sequence of a globally idempotent finite semigroup without identity element is at least 2n. (This had been conjectured by J. Wiegold.)

An Eisenstein-like criterion is proved for power series with algebraic coefficients satisfying algebraic differential equations of a certain general kind. The proof is elementary and the result extends earlier results of Hurwitz, Pólya and Popken

For a G/G/l queueing system let Xt be the number of customers present at time t and Yt(Zt) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let τl be the time until the number of customers in the sustem is reduced from j to j – l, given that X0 = j ≧ l, Y0 = y, Z0 = z. For the joint distribution of τl and Yτl and the Laplace transforms of the τl intergral equations are derived. Under slight conditions these integral equations have unique solutions which can be determined by standard methods. Our results offer a method for calculating the busy period distribution which is completely different from the usual fluctuatuion theoretic approach.

We study the characteristic p analogue of M-groups, the so-called Mp-group Generalizing this notion, we also consider the condition that the modular irreducible representations are induced from representations of dimension < p, or even weaker, of dimension not divisible by p.