In [1], R. L. Goodstein has extended some well-known theorems on functions and equations in a Boolean algebra to the case of a distributive lattice L with 0 and 1. The purpose of this paper is to prove that most of Goodstein's theorems, as well as some additional results, are still valid in the case when L is not required to have least and greatest elements.

We shall extend some of the results of (7) to the case of multiple alleles, our primary concern being that of polyploidy combined with multiple alleles. Generalisations often tend to make the computations more involved as is expected. Fortunately here, the attempt to generalise has led to a new method which not only handles the case of multiple alleles, but is an improvement over the method used in (7) for the special case of polyploidy with two alleles. This method which consists essentially of expressing certain elements of the algebra in a so-called “ factored ” form, gives greater insight into the structure of a polyploidy algebra, and avoids a great deal of the computation with binomial coefficients, e.g. see (7), p. 46.

Sr denotes an infinite space of r dimensions. Such a space is divided into two regions by a Sr−1. An infinite line is divided into n + 1 regions by n points. An infinite plane or other surface topically equivalent to it is divided into two regions by an infinite line.

§ 1. Corresponding to any partition

which we denote by [a1, . …, ak] or briefly by [a], of the integer n, we can construct a shape which has a1 spaces in the first row, a2 in the second row, . …, ak in the kth and last row. Thus the shape corresponding to the partition [5, 3, 3, 2] of 13 has the form:

The properties of the functions Pn, Qn may also be determined very briefly as follows:—

Let u be a function of x; then by integration by parts we have

By the work of Taskinen (see [4, 5]), we know that there is a Fréchet space E such that Lb(E, l2) is not a (DF)-space. Moreover there is a Fréchet–Montel space F such that is not (DF). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a (gDF)-space (cf. [1, Ch. 12 or 3, Ch. 8]). The (gDF)-spaces were introduced by several authors to extend the (DF)-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

The linear differential system

where w is a vector with n components and A is an n by n matrix is said to have z = 0 as a regular singular point if there exists a fundamental matrix of the form

such that S is holomorphic at z = 0 and R is a constant matrix ((1), p. 111; (2), p. 73). For such systems A will have at most a pole at z = 0 and we may write

where p is an integer, Ã is holomorphic at z = 0, and Ã(0) ≠ 0. However, the converse is not true. When p ≦ − 1, A is holomorphic at z = 0, and every fundamental matrix is holomorphic at z = 0. If p ≧ 1, the non-negative integer p is called (after Poincaré) the rank of the singularity and there is a significant difference between the cases p = 0 and p ≧ 1. If p = 0 the linear differential system (1) is known to have z = 0 as a regular singular point ((1), p. 111) ; whereas, if p ≧ 1, z = 0 may or may not be a regular singular point.

This is the problem discussed in my paper bearing the not very happy title “On the different non-linear arrangements of eight men on a chess-board”, which was read to the Edinburgh Mathematical Society on 14th March 1890, and is printed in its Transactions, Vol. VIII, p. 30. At that time I was not aware that the problem had been discussed by any previous writer, and I treated it as an entirely new one. I have since learnt that a good deal has been written about it, and I propose on the present occasion to give briefly the history of the problem, and the results which have been arrived at; also to communicate some new results which I have obtained.

The determination of the harmonic functions of elliptic and hyperbolic cylinders depends on the solution of Mathieu's differential equation. This equation, it has been remarked by Professor Whittaker, is the one which naturally comes up for study after the hypergeometric equation has been disposed of. Its solution presents difficulties which do not arise in connection with the hypergeometric equation or its degenerate cases, and it cannot, I think, be said that any discussion of the equation has yet been given with which the student of analysis can rest content. The treatment given below, though certainly incomplete at some points, seetns to follow the lines along which a thoroughly successful theory may be hoped for.