A homomorphism f: R → T of (commutative) rings is said to be universally incomparable in case each base change R → S induces an incomparable map S → S⊗RT. The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism f: R → T is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, k(f-1(Q))→k(Q), for each prime ideal Q of T. In several cases (f algebra-finite, T generated as R-algebra by primitive elements, T an overring of a one-dimensional Noetherian domain R), each universally incomparable map is shown to factor as a composite of an integral map and a special kind of radiciel.

Let F be a field. If A is an F-algebra with involution that is generated (as a space) by symmetric idempotents, then A is a subdirect product of copies of F if and only if every idempotent in A is symmetric.

Every group is the automorphism group of an orthomodular lattice.

We construct a counterexample, for p = 1, to the conjecture posed by Milaszevitch in 1970: is the space of functions which are analytic in the upper half plane and uniformly almost periodic in its closure dense in the Hardy space Hp (0 < p ∞) of analytic almost periodic functions?

Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then

(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),

(iv) if D corresponds to the shift operator (dn ≡ l), then .

Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

In J. Math. Anal. Appl. 61 (1977), 466–474, Leader has given a fixed point principle for an operator f:X → X, where X is a metric-space, based on conditional uniform equivalence of orbits. We generalize this principle for two mappings f1 and f2 to give common fixed point results in two different ways. Further we derive an f-generalized fixed point theorem for two commuting mappings.

We show that recent integral versions of the classic Jordan-Von Neumann characterization of Euclidean space may he viewed as special cases of a general averaging principle over sets of isometries.

Call a relation R⊆A2 (A some non-empty set) monadically representable when there exist F, G⊆A such that R = {(x, y) | x ∈ F ∘ y ∈ G} for some truth-functional connective ∘. This note finds a first-order condition on R which is necessary and sufficient for R to be monadically representable.

Sufficient conditions are derived for all bounded solutions of a class of linear systems of differential difference equations to be oscillatory.

An eulerian chain in a graph is a continuous route which traces every edge exactly once. It may be finite or infinite, and may have 0,1 or 2 end vertices. For each kind of eulerian chain, there is a characterization of the graphs which admit such a tracing. This paper derives a uniform characterization of graphs with an eulerian chain, regardless of the kind of chain. Relationships between the edge complements of various kinds of finite subgraphs are also investigated, and hence a sharpened version of the eulerian chain characterization is derived.

The method of proof of Magnus introduced in 1930 is adapted to prove the following theorem of Howie. If A and B are groups for which every finitely generated subgroup has an infinite cyclic image, and if one adds an additional relation (with obvious exceptions), then in the resultant group both A and B appear isomorphically.

In this paper we show that an R-module graph over a semisimple ring R can be written as a direct sum of graphic submodules that are uniquely determined up to isomorphism type. Moreover, this decomposition enables us to describe the R-module graph in graphic terms as a disjoint union of connected components, each of which consists of a complete directed graph on its vertices together with a set of loops at each vertex, determined by the loops at 0. We also give a graphic version of Maschke's Theorem.

Theorem 1 and Corollary 1 of the original paper [Bull. Austral. Math. Soc. 25 (1982), 99–101] are false unless, for example, we assume the Generalised Continuum Hypothesis. The proof of Theorem 1 correctly shows that exp(|c|) ≤ exp(|D|) but one cannot then deduce that the cardinality of C is at most that of D. All one can deduce is that the cardinality of C is less than exp(|D|).