In [6] we announced the followingConjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then

(a) F is a closed completely simple subsemigroup.

(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.

(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Let α be an irrational algebraic number of degree k over the rationals. Let K denote the field generated by α over the rationals and let a denote the ideal denominator of α. Then Cassels [3] has shown that for sufficiently large integral N > 0 distinctly more than half the integers n, are such that (n+α)a is divisible by a prime ideal pn which does not divide (m+α)a for any integer m ≠ n satisfying . The purpose of this note is to point out that minor modification of Cassel's proof enables the extension of the interval for n from to , and to derive results on the proportion of values n, for which the values f(n) of a given integral polynomial in n are divisible by a prime p > N.

Given a group G and a partial automorphism μ of G, i.e. an isomorphism mapping a subgroup A of G onto another subgroup B of G, then it is known [3] that μ can always be extended to a total automorphism, in fact an inner one, of a supergroup of G; that is there exists a group G* ⊇ G with an inner automorphism μ* whose effect on the elements of A is the same as that of μ. Also any number of partial automorphisms μσ, where a ranges over some index set Σ can be simultaneously extended to inner automorphisms of one and the same group [3, Theorem II].

R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then(i) R is finitely generated(ii) R is left alehra noetherian(iii) R is algebraic.Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems:Theorem 1. If there exist(i) a skew-Hadamard matrix H = U+I of order h,(ii)m-type matrices M = W+I and N = NT of order m,(iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying(a) XYT, YZT and ZXT all symmetric, and(b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

Repeated improper Riemann integrals arise in a variety of contexts, and the validity of changing the order of integration is often in question. Fubini's theorem ensures the equality of two repeated Lebesgue integrals when one of them is absolutely convergent. For many years I have assumed that an analogous test is applicable to repeated improper R-integrals, since they will be absolutely convergent and therefore in agreement with the corresponding L-integrals.

G(n;l) will denote a graph of n vertices and l edges. Let f0(n, k) be the smallest integer such that there is a G (n;f0(n, k)) in which for every set of k vertices there is a vertex joined to each of these. Thus for example fo = 3 since in a triangle each pair of vertices is joined to a third. It can readily be checked that fo = 5 (the extremal graph consists of a complete 4-gon with one edge removed). In general we will prove: Let n > k, andthen f0(n, k) = f(n, k).

In a recent paper, Z. Janko [5] announced the discovery of two new finite non-abelian simple groups and characterized these groups in terms of the centralizer of an involution. In fact, he proved the following result.

The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

The quadrature formula of order m using values of derivatives up to the m — 1st order with the best possible bound in is derived. Using certain properties of the polynomials minimal in Lq norm, it is proved that the optimal formula not use the derivatives of m — 1st order if m is even.

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

Let be a family of sets, {Fa|α∈Λ}. By the graph G of the system , we mean the graph whose set of vertices is and in which the vertices Fβ∈, are adjacent (that is, are joined by an edge) if and only if Fα ≠ Fβ and Fα ∩ Fβ □, where □denotes the empty set.

Let S be a poset with a greatest element 1. We denote order in S by ‘≦’ and, whenever they exist in S, l.u.b and g.l.b by ‘∨’ and ‘∧’ respectively. An orthocomplementation of S is a bijection w: S → S such that x ∨ xω exists for each x in S and (i) xωω = x, (ii) x ≦ y implies yω ≦ xω and (iii) x ∨xω = 1. If a poset S admits an orthocomplementation ω we call the pair (S, ω) an orthoposet.

In recent years a great deal of attention has been devoted to the study of finite simple groups, but one aspect which seems to have been little considered is that of the laws they satisfy. In a recent paper [3], the first two of the present authors gave a basis for laws of PSL(2, 5). The techniques of [3] can be used to show that (modulo certain classification problems) a basis for the laws of PSL(2, pn) can be made up from laws of the following types:

(1) an exponent law,

(2) laws which determine the Sylow subgroups,

(3) laws which determine the normalisers of the Sylow subgroups,

(4) in certain special cases, laws which determine subvarieties of smaller exponent, e.g. the subvariety of exponent 12 for those PSL(2, pn) which contain S4,

(5) a law implying local finiteness.

Let Tn denote a tree with n(≧ 2) labelled points: we assume Tn is rooted at a given point x, say the point labelled 1 (see [3] for definitions not given here). If we remove some edge e of Tn, then Tn falls into two subtrees one of which, say Tk, contains the root x. If k ≧ 2 we can remove some edge of Tk and obtain an even smaller subtree of Tn that contains x. If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ(Tn) denote the number of edges removed from Tn before the root x is isolated. Our main object here is to determine the expected value μ(n) and variance σ2(n) of λ(Tn) under the assumptions (1) Tn is chosen at random from the set of nn−2 trees with n labelled points that are rooted at point x, and (2) at each stage the edge removed is chosen at random from the edges of the remaining subree containing x. It follows from our results that μ(n) ~ (½πn)½ and (2−½π)n ~ (2−½π)n as n tends to infinity. We also consider the corresponding problem for forests of rooted trees and for trees in which the degree of the root is specified. We are indebted to Professor Alistair Lachlan for suggesting the original problem to us.

In [4] S. Ulam asks the following question.‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.

Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property:(1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;

there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.

U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].

It is the purpose of this note to illustrate by example the results obtained in Finch [1]. All our definitions and notation are as in that paper, and we use them without reference. In his paper, Finch considered orthoposets and, in particular, orthomodular posets that had either the A-property or the B-property.

Any congruence on a semigroup S with a nonempty set Es of idempotents induces a partition of the set Es. Two congruences ρ and σ on the semigroup S are defined to be idempotent-equivalent congruences on S if ρ and σ induce the same partition of Es. In this paper we investigate idempotent-equivalent congruences on orthodox semigroups (regular semigroups in which the set of idempotents forms a subsemigroup).

Free ideal rings (or firs, cf. [2, 3] and § 2 below) form a noncommutative analogue of principal ideal domains, to which they reduce in the commutative case, and in [3] a category TR of right R-modules was defined, over any fir R, which forms an analogue of finitely generated torsion modules. The category TR was shown to be abelian, and all its objects have finite composition length; more over, the corresponding category RT of left R-modules is dual to TR.

We consider the cosine functional equation (see [1, 2, 3]) , where f(z) is an entire function of a complex variable z and x, y are complex variables.