In this note we study commutative Baer rings, uniting the abstract algebraic approach with the approach of [3] using minimal prime ideals. Some new characterisations of this class of rings are obtained, relations between the minimal prime ideals of a commutative Baer ring B and its algebra EB of idempotents are considered, and some results concerning the direct decomposition of commutative Baer rings are given. We then study Baer ideals, and finally state without proof a new construction of the Baer extension of a commutative semiprime ring.

nHr denotes the number of n × n matrices with non-negative integral entries, with row-sums and column-sums all equal to r. Kenji Mano [3] investigated the number nHr in which n distinct objects each replicated r times can be distributed in equal numbers among n cells. He gives an intricate formula for the case r = 2. Recently, Anand et al. [1], making use of partitions, extended it to 3Hr and stated a plausible formula for nHr.

Given a function T mapping a Hausdorff locally convex topological vector space Φ into Φ and a point φ0 of Φ, convergence of the elementary filter associated with the sequence of iterates determined by T and φ0 is investigated. Sufficient conditions that the limit φ if it exists, be a fixed point of T are given and in the case Φ is the space of real valued functions of a real variable differentiability of the limit function φ is investigated.

Kato [3] has introduced a class of operators called strictly singular operators. These operators have many properties in common with compact operators. In fact the concept of a strictly singular operator is an extension of the concept of a compact operator. Kato has proved that if X and X′ are Banach Spaces, then the singular operators of X into X′ forms a closed subspace of the space of bounded linear operators of X into X′ and if X = X′, then these operators forms a two-sided ideal in the ring of bounded linear operators on X. He has also shown that the Riers-Schauder theorem holds for the spectrum of a strictly singular operator. Gohberg, Feldman and Markus [23] have treated the same class of operators with an equivalent definition.

Let B be a Banach space and let ℒ(B) denote the space of all bounded inear operators from B to B, which is a Banach algebra under composition of operators as multiplication. By a semigroup of operators G on B, we mean a norm bounded subser G of ℒ (B) which is a subsemigroup in the multiplicative structure of ℒ(B). The purpose of this paper is to study the existence of nonzero continuous linear functionals on B invariant under G, that is given B and G, does there exist μ∈B*, with μ ≠ 0, such that μ(Sx) = μ(x) for all x∈B and S∈G. This question is an attempt to generalize the familiar concepts of invariant means and amenability of semigroups. If H is any semigroup and m(H) is the Banach space of all bounded real valued functions on H with supremum norm, then a mean is a positive normalized continuous linear functional on m(H). A mean is called (left) [right] invariant if it is invariant under (left) [right] translations and H is called (left)[right] amenable if there exists such a mean; e.g., F is a left invariant mean if F∈m(H)* is such that (i) ‖F‖ = 1, (ii) F(x)≧0 if x(g)≧0 for all g∈H, (iii) F(xg) = F(x) for all x∈m(H) and g∈G where xg(h) = x(gh) for all h∈H. Amenability of semigroups has been studied extensively in recent years, for example see Day [2] or Hewitt and Ross [6] for an introduction, and Day [3] for a comprehensive survey.

Let E be a vector lattice in the sense of [2]. Two elements a, b in E are said to be disjoint (written a⊥b) if ′a′^′b′ = 0. For any subset A of E, A⊥ is the set If A is a one point set {a}, then we shall use a⊥ for {a}⊥. A⊥⊥ is denned by A⊥⊥ = {A⊥}⊥. A subset B of E is said to be a polar if it is of the form A⊥ for some subset A of E. It is well known [1] that the set of all polars of E ordered by inclusion forms a Boolean algebra. We shall denote this algebra by B, and M will mean the set of all maximal ideals of B. The following properties of polars will be used in the paper.

This paper is a continuation of [2] and will use the terminology and notation introduced therein. Basically, we seek to investigate the structure of indexed lattices in which the set of lengths of the minimal indexing chains is bounded.

Results concerning classes of null sets have been obtained by various authors. See, for example, [3], [4], [6], [7]. This paper contains results concerning classes of null sets and the notion of a ‘small system’. The motivation for considering ‘small systems’ comes from a paper by Riečan (c.f. [2]).

The main purpose of this paper is to prove that every automorphism of the semigroup of all Hadamard-differentiable mappings of a separable real Banach space into itself is inner. This generalizes the results of [7] which is a generalization of a result proved by Magill, Jr. [5].

Let H denote a set with three elements, and T3 the full transformation semigroup on X, i.e. T3 consists of the twenty-seven self maps of X under functional composition. A transformation semigroup (briefly a τ-semigroup) on three letters is an ordered pair (X, S), where S is any subsemigroup of T3.

Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

This is a continuation of our work [4]. The purpose of this paper is to find the relation of the class of improjective operators on some Banach space, with the classes of strictly singular, strictly cosingular and φ-admissible perturbations on X and to investigate those pairs of Banach spaces for which all bounded linear operators having either of the pair as domain and the other as range are improjective, or strictly singular. The beginning of Section 1 is intended to familiarize the reader with the concepts and notations used in this paper.

A group is called an s-group if it is locally finite and all its Sylow subgroups are abelian. Kovács [4] has shown that, for any positive integer e, the class se of all s-groups of exponent dividing e is a (locally finite) variety. The proof of this relies on the fact that, for any e, there are only finitely many (isomorphism classes of) non-abelian finite simple groups in se; and this is a consequence of deep results of Walter and others (see [6]). In [2], Christensen raised the finite basis question for the laws of the varieties se. It is easy to establish the finite basis property for an se which contains no non-abelian finite simple group; and Christensen gave a finite basis for the laws of the variety s30, whose only non-abelian finite simple group is PSL(2,5). Here we prove Theorem For any positive integer e, the varietysehas a finite basis for its laws.

A generalization of Neumann's integral connecting the two kinds of Legendre function is obtained. It contains an extra parameter which is not a function of the parameters of the Legendre functions, unlike all previous extensions of the original formula. These extensions are shown to be particular cases of the new generalization and some further particular cases are also indicated.

The work in this paper is directed at the question: What differentiable flows on [0, 1] [1] are conjugates of linear fractional flows on [0, 1]?

Let IE, ∧ and ∧R denote the collections of all non-negative integers, isols and regressive isols respectively. An ω-group is a pair (α, p) where (1) α ⊆ E, (2) p(x, y) is a partial recursive group multiplication for α and (3) the function which maps each element of α to its inverse under p has a partial recursive extension. If G = (α, p) is an ω-group, we call the recursive equivalence type of a the RET or order of G (written o(G)). Let GR = {T∈∧R′T = o(G) for some ω-group G}. It follows from the version of the Lagrange Theorem given in [4] that ∧R − GR is non-empty and has cardinality c. In this paper we characterise the isols in GR as follows: A regressive isol T belongs to GR if and only if T∈E or T is infinite and there exist a regressive isol U ≦ T and a function an from E into E − {0} such that U ≦*an and T = ΠUan. (The “≦*” is denned in [2]). In presenting the proof of this result, we shall assume that the reader is familiar with either [3] or [4]. The proof that, given an and U ≦*an, a group of order ΠUan exists is based on the natural trick—one constructs a direct product of disjoint cyclic groups of order a0, a1,… indexed by elements of a set of RET U. The proof that any regressive group G has order of the form ΠUan is trivial for finite groups; the proof for infinite regressive groups is based upon the construction of an ascending chain of finite subgroups Gi of G such that and .

Dr. Roy O. Davies, of the University of Leicester, has sent me a neat proof of Theorem 3, p. 430, of my paper “Changing the Order of Integration” [1]. Perhaps other readers would appreciate this improvement, as I do; I therefore present it below, with his agreement, together with a consequent improved proof of Theorem 2. The counter-example, which is the main part of that paper, is unaffected.