A necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

The generating function for the number of linear partitions was found by Euler, the method being almost trivial. That for plane partitions is due to Macmahon, but, even in a simplified form found by Chaundy, the proof is far from trivial. The number of solid partitions of n, i.e. the number of solutions of

is denoted by r(n). It has often been conjectured that the generating function of r(n) is , but this is now known to be false. We write η(a, b, c) for the generating function of the number of solutions of (1) subject to the additional condition that

Macmahon 1916 found n(a, 1, 1) for general a. Here we find η(a, b, c) for general a, b. c.

Throughout this paper we shall be concerned with n × n matrices X=[xij] whose elements xij belong to a given Boolean algebra B(≤, ∩, ∪, ′). In tne first part of the paper we show that the set S(λ, x) of matrices with a given eigenvector x and eigenvalue λ is a subgerbier (i.e.), γ-semi-reticulated sub-semigroup of Mn(B), the Boolean algebra of all n × n matrices over B. We also determine the greatest element M(λ, x) of S(λ, x) and consider some of its properties. In the second part of the paper we consider the structure of matrices which possess a given primitive eigenvector x and show in particular that, for any given λ ∈ B, there is a matrix, namely M(λ, x), having x as the maximum primitive eigenvector associated with the eigenvalue λ. We also determine necessary and sufficient conditions under which the matrix M(λ, x) is the same for all λ in any given sublattice of B.

A multiple integral, whose integrand is an n × n determinant, is evaluated over certain regions of n-dimensional space. Similar integrals are encountered in the theory of Zonal polynomials. In the course of the work a partition problem arises. In the next paper of these Proceedings, Professor Rutherford enumerates these partitions and relates the subject to the theory of the representation of the symmetric group.

§ 1. The principal object of this note is to establish formula (16) of the preceding paper by H. Jack (1966). This formula, which was conjectured by Jack, evaluates a certain coefficient which is attached to a symbol {p, q, …, r}. In this symbols, p, q, …, r form a partition of m such that o≤p≤q≤…≤r,p+q+…+r=m. The symbol however vanishes if any two of the integers p, q, … r are equal but non-zero. In the remaining cases we have to show that the coefficient in question has the value

The theory of Hypo-Elasticity is generalized so as to include the theory of (anisotropic) elasticity as a special case, and to include thermal effects. Explicit restrictions on the constitutive equations, arising from thermodynamics, are obtained.

Necessary and sufficient conditions are given for the existence of solutions of equations, in any number of variables, on distributive lattices with least and greatest elements, together with an algorithm for determining a solution when these conditions are satisfied.