It has been shown ([8], [2], [1], [3], [9]) that a collection of physical operations or experiments can be represented by a nonempty set of nonempty sets satisfying certain technical conditions. Such a set is called a manual. The operations in are looked at as having no “before” and no “after”, i.e., they are isolated in time. If we wish to look at connected sequences of operations—in particular, if we wish to condition by events in —we must look at the compound manual c whose elements represent compound operations built up from the operations in the base manual .

In [12], Loy and Miller proved that a locally compact, eudoxian TR group is algebraically and order-theoretically (and hence, topologically) isomorphic to a finite product of copies of the real numbers. In [18], Wirth used their result to describe the subgroup of a locally compact TR group generated by the compact neighbourhoods of zero. The proof of Loy and Miller relied heavily on a result of Mackey (cf. [10], p. 390) and either the finite-dimensional case of the Choquet-Kendall Theorem (cf. [15], pp. 9–10) or the representation theory of Kakutani (cf. [11], Appendix). Below we use only elementary topological results and order-theoretic arguments and a theorem of Conrad [4] to characterize all non-secular, locally compact TRL groups (Theorem 3). Our proof of Theorem 3 allows us to deduce algebraically the theorems both of Loy and Miller and of Wirth, in both cases without appealing to the theorem of Conrad.

There is a large body of literature on inverse semigroups. This literature contains a considerable amount of information concerning congruences on these semigroups, which is not surprising in view of the demonstrated fact that congruences on inverse semigroups play a decisive role in most of the existing structure theorems. In addition, for an inverse semigroup of known structure, finding its congruence lattice, or even certain properties of this lattice, often gives information about these semigroups not apparent in their structure theorems.

Let K be a non-abelian cubic field of discriminant D, and ζK(s) its Dedekind zeta-function. Set ψ(s) = ζk(s)/ζ(s). Then it is known that ψ(s) is the Artin L-series associated with the field K. It is also known that ψ(s) is an entire function of order 1.

In his book Transversal Theory [3], L. Mirsky has remarked that “At present, the relation between Dilworth's decomposition theorem…and transversal theory is rather tenuous; but further study may reveal unexpected connections”. Some of these connections can perhaps now be seen a little more clearly; and our purpose in this note is to make one or two observations in this regard. Throughout, all sets considered are finite.

Over the last few years, various extensions of the topological degree of a mapping have been made so as to include non-compact perturbations of the identity. One such extension, which employs compactness conditions, has been to the class of limit compact maps which were extensively studied by Sadovsky [7]. The class is a large one as it contains all compact mappings, contraction mappings and, more generally, condensing mappings. Sadovsky [7] gives a theory of degree for maps of the form I-f, where f is limit compact, and this was extended independently and with different methods by Petryshyn and Fitzpatrick [4] and the author [9] to allow f to be a multi-valued mapping. A refinement of the methods of [9] was given by Vanderbauwhede [8].

It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

If R is a ring and S is a semigroup, the corresponding semigroup ring is denoted by R[S]. A ring is semiprime if it has no nonzero nilpotent ideals. A semigroup S is a semilattice P of semigroups Sα if there exists a homomorphism φ of S onto the semilattice P such that Sα = αφ−1 for each α ∈ P.

Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let Rbe a ring with 1. If His finite and G is not finitely generated we show that any non–zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.

A semilattice of cancellative semigroups S is a p.o. semigroup with the order relation a ≤ b iff ab = a2. If S is a strong semilattice of cancellative semigroups (i.e., multiplication in S is given by structure maps ϕe,f (f ≤ e in E)), for each supremum-preserving completion Ē of the semilattice E there is a strong semilattice of cancellative semigroups T over Ē which is a supremum-preserving completion of S in ≤. Given Ē, T is constructed directly. In this paper it is shown that multiplication by an element of S distributes over suprema in ≤ if E has this property (called strong distributivity). Next it is shown that the completion construction also applies to a semilattice of cancellative semigroups which is not strong if S is commutative and Ē is strongly distributive. Finally, it is shown that for semilattices of cancellative monoids a completion is completely determined, up to isomorphism over S, by completions of E.

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.

Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t < p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.

1. Let X be a closed Riemann surface of genus g ≥ 2 and let Aut X denote the group of automorphisms of X where, in this paper, an automorphism means a conformal or anticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformal automorphism J of order 2 such that X/H has genus 0, where H = 〈J〉 is the group of order 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branched over 2g + 2 points and J is the sheet-interchange map. J is the unique conformal automorphism of order 2 such that X/〈J〉 has genus 0 and it follows that if U ∈ Aut X, then UJU−1 = J. Thus J is central in Aut X and H ≤ Aut X. (Cf. [8])

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a → ba, a → ab, a → ab + ba, a → bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms of which are weakly compact perturbations of the identity.

Suppose that m0 is an integer, m0≥3, ρ = exp(2πi/m0), K = ℚ(ρ, i), v denotes the degree of K, ξ∈K has degree N over ℚ. The length, where is the (irreducible) minimal polynomial for with ξ relatively prime integer coefficients. Feldman [2, p. 49] proved that there is an absolute constant c0>0 such that

From [2, p. 49, Notes 1 and 2] we know that v = φ(m0) or v = 2φ(m0), and φ(m0)≥ c1m0(log log m0)−1 (c1 > an absolute constant), where φ(m0) denotes Euler's function.

The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p > 0, then kG is locally Noetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

The following theorem was proved in [1].

Theorem 1. Let S and T be continuous, commuting mappings of a complete, bounded metric space (X, d) into itself satisfying the inequality

for all x, y in X, where 0 ≤ c < 1 and p, p′, q, q′ ≥ 0 are fixed integers with p + p′, q + q′ ≥ 1. Then S and T have a unique common fixed point z. Further, if p′ or q′ = 0, then z is the unique fixed point of S and if p or q = 0, then z is the unique fixed point of T.

In [3] E. Bishop introduced the notion of an operator with a “duality theory of type 3” and gave a certain sufficient condition for an operator to have a duality theory of type 3. In this note we show that in fact Bishop's sufficient condition implies that a given operator is decomposable [4]. Moreover, this condition characterizes a decomposable operator.

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . For undefined terminologies used in the paper, we refer to [3] and [6].