The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.

In previous papers (Bartlett 1951, Williams 1952a, b, c; 1953, 1955) certain exact tests for discriminant functions have been discussed. These papers have also indicated how the tests may usefully be applied to a wide range of problems: the significance of discriminant function coefficients, the interpretation of interactions in factorial experiments, the concurrence of regression lines with the same independent variables and the proportionality of regression lines with different independent variables, the comparison of sets of ratios, and the testing of arbitrary scores applied to frequency data.

In the development of the rotor calculus presented in a previous paper space-time was taken to be flat. This work is now extended to the case of curved space-times, which, in the first instance, is taken to be Riemannian. (The calculus bears at times a strong formal resemblance to the spinor analysis of Infeld and van der Waerden, but it is in fact developed quite independently of this.) Owing to the fact that all general relations had earlier already been written in a generally covariant form they may be taken over unchanged into the present context. In particular,

now serves as a defining relation for the connecting quantities τAkl. A linear rotor connection is introduced, and the covariant derivative of a rotor defined. The covariant constancy of the τAkl establishes the relation between the linear connections in w-space and in r-space. A rotor curvature tensor is considered alongside a number of other curvature objects. Next, conformal transformations are dealt with, of which duality rotations may be considered as a special case. This leads naturally to a gauge-covariant generalization of the whole calculus. A so-called rotor-derivative is defined, and some general relations involving such derivatives investigated. The relation of the rotor curvature tensor to the spin-curvature tensor is touched upon, after which the introduction of “geodesic frames” is considered. After this general theory some points concerning the Maxwell field are dealt with, which is followed by some work concerning basic quadratic and cubic invariants. Finally, a certain basic symmetric rotor is re-considered in the context of the classification of Weyl tensors, and the idea of a canonical representation suggested.

An (r, 1) system is pair (V, F) where V is a v-set and F is a family of non-null subsets of V (b in number) which satisfy the following.

(i) Every pair of distinct members of V occur in precisely one member of F;

(ii) Every member of V occurs in precisely r members of F.

A pseudo parallel complement PPC(n, α) is an (n+1, 1) system with ν = n2 – αn and b ≦ n2 + n – α in which there are at least n – α blocks of size n. A pseudo intersection complement PIC (n, α) is an (n+1, 1) system with ν=n2 – αn + α – 1 b ≦ n2+n – α in which there are at least n – α+1 blocks of size n-1. It is shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α – 1) and that for n > (α4 + 2α3 + 2α2 + α)/2 every PPC(n, α) can be embedded in a projective plant of order n. The latter generalizes a result of Totten (who proves the result for α = 1).

It is shown that if all powers of a ring element a are regular, then a is strongly pi-regular exactly when a suitable word in the powers of a and their inner inverses is a unit.

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.

This paper considers (left) near-rings which satisfy the left self distributive (LSD) identity: abc = abac. This is exactly the class of near-rings for which each left multiplication mapping, τa: x → ax, is a near-ring endomorphism. Simple and subdirectly irreducible ones are classified and semidirect sum decompositions into reduced and nilpotent pieces are given. LSD near-rings with restrictive conditions on nilpotent elements or annihilating sets are considered. Type 1 prime (semiprime) ideals in an LSD near-ring are completely prime (semiprime). Further results on prime and maximal ideals are given. Numerous examples are given to illuminate the theory and to illustrate its limitations. Some analogous theory for right self distributive near-rings is given (those satisfying the identity: abc = acbc).

The systematic study of identities in groups was begun by B. H. Neumann in 1937, [7]. Although many special loop identities have been extensively investigated, there is not, as yet, a general theory of identical relations in loops. We cannot hope that such a theory will be as rich in results as in the group case (except in the sense that loop theory includes group theory) but there are many interesting problems, including some which have no analogue in the associative case.

It is proved that a regular essentially closed and weakly homomorphically closed proper subclass of rings consists of semiprime rings. A regular class M defines a supernilpotent upper radical if and only if M consists of semiprime rings and the essential cover Mk of M is contained in the semisimple class S U M. A regular essentially closed class M containing all semisimple prime rings, defines a special upper radical if and only if M satisfies condition (S): every M-ring is a subdirect sum of prime M-rings. Thus we obtained a characterization of semisimple classes of special radicals; a subclas S of rings is the semisimple class of a special radical if and only if S is regular, subdirectly closed, essentially closed, and satisfies condition (S). The results are valid for alternative rings too.

We here study permutational products of groups, a study which was initiated by B. H. Neumann [8], [9].

Certain integers have the property that they can be partitioned into distinct positive integers whose reciprocals sum to 1, e.g., and In this paper we prove that all integers exceeding 77 possess this property. This result can then be used to establish the more general theorem that for any positive rational numbers α and β there exists an integer r(α, β) such that any integer exceeding r(α, β) can be partitioned into distinct positive integers exceeding β whose reciprocals sum to α.

Let A be an Archimedean, uniformly complete, semiprime f-algebra and F(X1,…Xn) ∈ R+ [X1,…Xn] a homogeneous polynomial of degree p (p∈ N). It is shown that (F(u 1…unn))1/p exists in A+ for all u1…un ∈ A+.

Introductory. This paper considers a canonical form, or rather a class of canonical forms, for three dimensional probability distributions subject to a rather mild restriction. These canonical forms are used to develop suitable tests of independence and lead to a consideration of the partition of χ2 in the analysis of complex contingency tables. Where these methods and Bartlett's are both applicable it is shown that they give comparable results; but the partitioning methods are more general.

We give several results concerning the simultaneous approximation of certain complex numbers. For instance, we give lower bounds for |a–ξo |+ | ea – ξ1 |, where a is any non-zero complex number, and ξ are two algebraic numbers. We also improve the estimate of the so-called Franklin Schneider theorem concerning | b – ξ | + | a – ξ | + | ab – ξ. We deduce these results from an estimate for linear forms in logarithms.

Two subgroups ME(G) and MI(G) of the Schur multiplier M(G) of a finite group G are introduced: ME(G) contains those cohomology classes [α] of M(G) for which every element of G is α-regular, and MI(G) consists of those cohomology classes of M(G) which contain a G-invariant cocycle. It is then shown that under suitable circumstances, such as when G has odd order, that each element of MI(G) can be expressed as the product of an element of ME(G) and an element of the image of the inflation homomorphism from M(G/G′) into M(G).

In this article, we prove that for a finite field Fq with even q > 3, any complete mapping polynmial of Fq has reduced degree at most q − 3. This is a solution to a problem of Niederreiter and Robinson about finite fields.;

Throughout this paper, G denotes a Hausdorff locally compact Abelian group, X its character group, and Lp(G) (1 ≦ p ≦ ∞) the usual Lebesgue space formed relative to the Haar measure on G. If f ∈ Lp(G), we denote by Tp[f] the closure (or weak closure, if p = ∞) in Lp(G) of the set linear combinations of translates of f.

The stochastic birth-death process considered in this paper provides an approximate model for phage reproduction in a bacterium. In a recent paper, Hershey [1] has discussed reproduction and recombination in phage crosses, and a deterministic model for the reproductive process has been the subject of a previous note by the author [2]. A very readable account of the process is given by Sanders [3] in his recent article, “The life of viruses”.

In this note we investigate lacunarity or ‘thin’ subsets in the dual object of a compact group via different classes of summing operators between Banach spaces. In particular, we give characterisations of Sidon and ∧ (p) sets, 2 < p < ∞