Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so that

Assume also that

Let A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.

In the following we use the notation and terminology of [6] and [7]. If S is an inverse semigroup, then Es denotes the semilattice of idempotents of S. If a is any element of the inverse semigroup, then a−1 denotes the inverse of a in S. An inverse subsemigroup S of an inverse semigroup S′ is self-conjugate in S′ if for all x ∈ S′,x−1Sx ⊆ S; if this is the case, S′ is called a conjugate extension of S. An inverse subsemigroup S of S′ is said to be a full inverse subsemigroup of S′ if Es = Es′. If S is a full self-conjugate inverse subsemigroup of the inverse semigroup S′, then S is called a normal inverse subsemigroup of S′, or, S′ is called a normal extension of S.

The purpose of this note is to answer in the affirmative a long standing open question raised by Singal and Singal — whether every almost continuous function is closure continuous (θ-continuous).

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

There exist several different approaches to the problem of solving dual integral equations involving Bessel Functions [1, 2, 3, 4, 5, 6,7], and Erdelyi and Sneddon in a recent paper [8] have shown that the introduction of certain operators occurring in the theory of fractional integration enables the relationships between the various methods to be clearly demonstrated. For dual integral equations other than those involving Bessel Functions the operators introduced by Erdélyi and Sneddon are not always the appropriate ones to use and it seems to be of interest to consider this more general type of situation.

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 R be a commutative Noetherian ring and G a group of elements acting on R as automorphisms. In this note, we are concerned with the structure of the lattice of invariant ideals of R. In particular we shall compute the Krull dimension of this lattice. Our group is an arbitrary group. There are none of the usual assumptions of some sort of algebraic action.

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local form

The restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.

This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

The main object of this note is to show that a proof given by A. J. Macintyre [2] of a result on the overconvergence of partial sums of power series works more easily in the context of Dirichlet series. Applying this observation to the particular Dirichlet series Σane−ns, we can remove certain restrictions which Macintyre finds necessary in the direct treatment of power series.

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.

Let N be a homomorphically closed class of associative rings. Put N1 = Nl = N and, for ordinals a ≥ 2, define Nα (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class IN (lower left strong radical class IsN) determined by N. The chain {Nα} is called Kurosh's chain of N. Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ Nk for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = Nn+l ≠ Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classes N with Nα = Nα for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that Nα = Nα for all α and Nn ≠ Nn+i = Nn+2. This in particular answers Question 6 of [4].

B. Rouxel [7] and S. J. Li and C. S. Houh [6] have generalised the notion of an -submanifold (Chen submanifold) to an k-submanifold. In [1] we have studied the relation between their definitions for the Euclidean case.

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.