A.H. Clifford ([2], [3]) has shown that all finite dimensional irreducible representations of a completely 0-simple semigroup can be obtained as extensions of those of its maximal subgroups. Lallement and Petrich, [7], have given an alternative method for constructing the irreducible representations of a finite 0-simple semigroup from its Schutzenberger representation ([13]). Using the form which they obtain for the irreducible representations of a finite 0-simple semigroup S = M0(G; m, n; P), Lallement and Petrich show that where M is the Munn algebra isomorphic to the (contracted) algebra Φ(S) of S over Φ.

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.

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G, must there be a minimal set of generators of G that contains an element whose order is divisible by p? A set of generators of G is minimal if no set with fewer elements generates G. A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

Let S be a nonempty subset of a normed linear space E. A point s0 of S is called a farthest point if for some x ∈ E, . The set of all farthest points of S will be denoted far (S). If S is compact, the continuity of distance from a point x of E implies that far (S) is nonempty.

In this paper we prove the existence of simplexes self-polar for a given simplex in projective space of dimensions higher than two. And noting the reciprocity of the relationship between a simplex and any simplex self-polar for it, further, we consider a configuration in projective space associated with a pair of mutually self-polar simplexes. Finally the results obtained are related to the simplexes determined by Hadamard matrices in space of special dimensions.

The purpose of this paper is to derive asymptotic relations giving the error of a Gauss type quadrature, applied to analytic functions, in terms of certain coefficients in the orthogonal expansion of the integrand. The Fourier expansions of the integrand we consider here are those in terms of the Legendre and the Chebyshev polynomials. In Section 3 we obtain the error of the Gauss-Legendre quadrature expressed in terms of the Legendre-Fourier coefficients of the integrand. In Section 4 the errors of Gauss-Legendre, Lobatto and Radau quadrature formulas are obtained, for large n, expressed in terms of the Chebyshev-Fourier coefficients of the integrand. In deriving these estimates we have used complex variable methods restricting ourselves to the class of analytic integrands; this allows us to obtain simple contour integral representations for the errors of these quadratures for large values of n. However, the form of the estimates obtained indicate that these are applicable to a much wider class of functions.

A semigroup S is called regular if a ∈ aSa for every element a in S. The elementary properties of regular semigroups may be found in A. H. Clifford and G. B. Preston [1]. A semigroup S is called orthodox if S is regular and if the idempotents of S form a subsemigroup of S.

An operator acting on a Banach space is said to be unicellular if its lattice of invariant subspaces is totally ordered by inclusion. Each weighted shift on a sequence space has a natural totally ordered set of invariant subspaces, and is unicellular if these are its only invariant subspaces.

Let F be an algebraic number field, and S a subgroup of the general linear group GL(n, F). We shall call S a U-group if S satisfies the condition (U): Every x ∈ S is a matrix all of whose eigenvalues are algebraic integers. (This is equivalent to either of the following conditions: a) the eigenvalues of each matrix (x are all units as algebraic numbers; b) the characteristic polynomial for x has all its coefficients integers in F. In particular, then, every group of matrices with entries in the integers of F is a U-group.

If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

A (v, k, λ) configurations is conjectured to exist for every v, k and λ satisfying λ(v − 1) = k(k − 1) and k − λ is a square if v is even, x2 = (k − λ)y2 + (−1)(v−1)/2 λz2 has a solution in integers x, y and z not all zero for v odd.