Let be a commutative Banach algebra over the complex field C, M an ideal of . Denote by M2 the set of all finite linear combinations of products of elements from M. M will be termed idempotent if M2 = M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

A variety of groups is an equationally defined class of groups: equivalently, it is a class of groups closed under the operations of taking cartesian products, subgroups, and quotient groups. If and are varieties, then is the class of all groups G with a normal subgroup N in such that G/N is in ; is a variety, called the product of and . We denote by the variety generated by the unit group, and by the variety of all groups. We say that a variety is indecomposable if , and cannot be written as a product , with both and One of the basic results in the theory of varieties of groups is that the set of varieties, excluding , and with multiplication of varieties as above, is a free semi-group, freely generated by the indecomposable varieties. Thus one would like to be able to decide whether a given variety is indecomposable or not. In connection with this question, Hanna Neumann raises the following problem (as part of Problem 7 in her book [7]): Problem 1. Ifandprove that [] is indecomposable unless bothandhave a common non-trivial right hand factor.

Let ω(x) be non-decreasing on the closed interval [a, b]. Outside the interval ω(x) is defined by ω(x) = ω(a) for x < a and ω(x) = ω(b) for x > b. Let S denote the set of points of continuity of ω(x) and D denote the set of points of discontinuity of ω(x). R. L. Jeffery [5] has defined the class U, of functions f(x) as follows:

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a rational polygon?’ Many of the known results concerning this question are contained as special cases in theorem 1 below which was proved by one of us (cf. the references).

To prove the statement given in the title take a set Σ1 of identities characterizing distributive lattices 〈L; ∨, ∧, 0, 1〉 with 0 and 1, and let Then is Σ redundant set of identities characterizing Stone algebras = 〈L; ∨, ∧, *, 0, 1〉. To show that we only have to verify that for a ∈ L, a* is the pseudo-complement of a. Indeed, a ∧ a* 0; now, if a ∧ x = 0, then a* ∨ x* 0* = 1, and a** ∧ = 1* = 0; since a** is the complement of a*, the last identity implies x** ≦ a*, thus x ≦ x** ≦ a*, which was to be proved.

Problem 24 of Hanna Neumann's book [3] reads: Does there exist, for a given integer n > 0, a Cross variety that is generated by its k-generator groups and contains (k+n)-generator critical groups? In such a variety, is every critical group that needs more than k generators a factor of a k-generator critical group, or at least of the free group of rank k? In a recent paper [1], R. G. Burns pointed out that the answer to the first question is an easy affirmative, and asked instead the question which presumably was intended: Given two positive integers k, l, does there exist a variety 23 generated by k-generator groups and also by a set S of critical groups such that S contains a group G minimally generated by k+l elements and S/{G} does not generate B? The purpose of this note is to record a simple example which shows that the answer to the question of Burns is affirmative at least for k = 2, l = 1, and also that the answer to the second question of Hanna Neumann's Problem 24 is negative.

Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.

Let S be a bisimple semigroup and let Es denote its set of idempotents. We may partially order Es in the following manner: if e, f ∈ E s, e ≧ f if and only if ef = fe = e. We then say that Es is under or assumes its natural order. Let I0 denote the non-negative integers and let n denote a natural number. If Es, under its natural order, isomorphic to (I0)n under the reverse of the usual lexicographic order, we call S an ωn-bisimple semigroup. (See [9] for an explanation of notation.) We determined the structure of ωn-bisimple semigroups completely mod groups in [9]. The ωn-bisimple semigroups, the I-bisimple semigroups [8], and the ωnI-bisimple semigroups [9] are classes of simple semigroups except completely simple semigroups whose structure has been determined mod groups.

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

Consider the following statement: For every positive integer n and every prime p there is a finite p-group of nilpotency class (precisely) c all of whose (n−1)-generator subgroups are nilpotent of class at most n.

Distributive pseudo-complemented lattices form an extensively studied class of distributive lattices. Examples are the lattice of all open sets of a topological space, the lattice of all ideals of a distributive lattice with zero and the lattice of all congruences of an arbitrary lattice. Lattice which are just pseudo-complemented have been studied in detail by J. Varlet [6], [7] where, however, the most interesting results require at least the assumption of modularity, sometimes distributivity.

Using the notion of spherical modification and results from Morse theory a general technique is described for constructing manifolds whose strong category is small (≦ 3) but whose homological structure is complex.

Let E be a real infinite-dimensional Banach space. Let ℒ be the Banach algebra of all continuous linear mappings of E into itself with topology defined by the norm:

We record here two further remarks about the systems, studied in [1] and [2], consisting of a vector space U and a set K of subspaces of U. In § 1, we show that such a system may be viewed as a module over a suitable artinian ring; the results of [1] and [2] thus serve to illustrate the complexity of structure of these modules. The main idea, a little wider than one introduced by Mitchell in Chapter IX of [3], is to view a diagram of vector spaces, with a small category as the scheme of the diagram, as a module over the ‘category ring’ of the category.

By the systematic use of Fourier transforms and suitable weight functions L. R. Volevich and B. P. Paneyakh brought many classes of spaces of distributions (including the Sobolev spaces) and their topological duals under the one unifying definition. The main purpose here is to demonstrate that the representation of multipliers between pairs of these spaces (that is, continuous linear operators from one space into another which commute with translations) may be related to the representation of multipliers between Lp and Lq.