An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.

Some new concepts are introduced, in particular that of a unique factorization semilattice. Necessary and sufficient conditions are given for two principal ideals of the semilattice of idempotents of a free inverse monoid FIM(X) to be isomorphic and some properties of the Munn semigroup of E[FIM(X)] are obtained. Some results on the embedding of semilattices in E[FIM(X)] are also obtained.

Let S denote a compact semitopological semigroup (i.e. the multiplication is separately continuous) and P(S) the set of probability measures on S. Then P(S) is a compact semitopological semigroup under convolution and the weak * topology (4). Let Γ be a subsemigroup of P(S) and where supp μ is the support of μ ∈P(S). In the case in which S is commutative it was shown by Glicksberg in (4) that S(Γ) is an algebraic group in S if Γ is an algebraic group. For a general semigroup S, Pym (7) considered Γ = {η}, η being an idempotent, and established that S(Γ) is a topologically simple subsemigroup of S, i.e. every ideal of S(Γ) is dense in S(Γ). In this note we prove that if Γ is a simple subsemigroup of P(S) (a semigroup is simple if it contains no proper ideal) which contains an idempotent then S(Γ) is a topologically simple subsemigroup of S. We also give an example to show that our conclusion (hence also Pym's) is best possible in the sense that S(Γ) is not simple in general

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

Relative uniform limits need not be unique in a non-archimedean partially ordered group, and order convergence need not imply metric convergence in a Banach lattice. We define a new type of convergence on partially ordered groups (R-convergence), which implies both the previous ones, and does not have these defects. Further R-convergence is equivalent to relative uniform convergence on divisible directed integrally closed partially ordered groups, and to order convergence on fully ordered groups.

The subjoined is an easy geometrical proof of the following theorem (which derives its importance from being part of the method of Chasles of constructing geometrically the ninth point when eight points of an “associated system” are given).

Theorem. If a point be taken on the radical axis of a coaxial system of circles, and from it tangents be drawn to any circle of the system, these tangents are cut in points on a conic, by the radical axis of the circle and a given fixed point. The two points are the foci of the conic. (Fig. 1.) Let W1W2 be the line of

gives an ellipse as the locus of P1P2, &c, when, as in Fig, 1, S is internal to F. If S were an external point, we should have P1F-P1S = P1F - P1f1 = radius of F = constant, and the locus of P1, P2, &c., would be a hyperbola. When F is at infinity on the radical axis, P1S = P1f1, and P1f1 being at right angles to W1W2, the conic is a parabola, and the line of centres the directrix.

Every distributive double p-algebra L is shown to have a congruence permutable extension K such that every congruence of L has a unique extension to K.

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

Let be an arbitrary semigroup. A congruence γ on is a group congruence if /γ is a group. The set of group congruences on is non-empty since × is a group congruence. The lattice of congruences on a semigroup will be denoted by () and the set of group congruences on will be denoted by (). If () is a lattice then it is modular and γ ∨ ρ = γ ο ρ = ρ ο γ for all γ, ρ ε (). The main result is that γ ν ρ = γ ο ρ ο γ for any γ ε () and ρ ε () (whence every element of the set () is dually right modular in (). This result has appeared, for particular classes of semigroups, many times in the literature. Also γ ν ρ = γ ο ρ ο γ = ρ ο γ ο ρ for all γ, ρ ε () which is similar to the well known result for the join of congruences on a group. Furthermore, if γ ∩ ρ ε () then γ ν ρ = γ ο ρ = ρ ο γ.

The two problems are :—

1. To divide a given straight line internally and externally so that the ratio between its segments may be equal to a given ratio.

2. To divide a given straight line internally and externally so that the rectangle under its segments may be equal to a given rectangle.