An inverse semigroup is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper [4], the author showed that any E-unitary inverse semigroup is isomorphic to a semigroup constructed from a triple (G, ℋ, ) consisting of a down-directed partially ordered set ℋ, an ideal and subsemilattice of ℋ and a group G acting on ℋ, on the left, by order automorphisms in such a way that ℋ = G. This semigroup is denoted by P(G, ℋ, ); it consists of all pairs (a, g)∈ × G such that g−1a ∈ , under the multiplication

Beineke and Schwenk [1] have defined the bipartite Ramsey number R(m,n), for integers m, n (1≤m≤n ), to be the smallest integer p such that any 2-colouring of the edges of the complete bipartite graph Kp, v forces the appearance of a monochromatic Km, n. In [1] the following results are established:

with equality if there is a Hadamard matrix of order 2(n−1), n odd,

if there is a Hadamard matrix of order 4(n−1),

In a recent paper [4] Bennett and Kalton characterized dense, barrelled subspaces of an arbitrary FK space, E. In this note, it is shown that if E is assumed to be an AK space, then the characterization assumes a simpler and more explicit form.

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

In a recent paper [5] Tits proves that a linear group over a field of characteristic zero is either solvable-by-finite or else contains a non-cyclic free subgroup. In this note we determine all the infinite irreducible solvable-by-finite subgroups of GL(2, F), where F is an algebraically closed field of characteristic zero. (Every reducible subgroup of GL(2, F) is metabelian.) In addition, we prove that an irreducible subgroup of GL(2, F) has an irreducible solvable-by-finite subgroup if and only if it contains an element of zero trace.

In this paper G is a nondiscrete compact abelian group with character group Г and M(G) the usual convolution algebra of Borel measures on G. We designate the following subspaces of M(G) employing the customary notations: Ma(G) those measures which are absolutely continuous with respect to Haar measure; MS(G) the space of measures concentrated on sets of Haar measure zero and Md(G) the discrete measures.

The spectrum of a prespectral operator was investigated by Dowson in [4]. The question was left open there whether, if a prespectral operator has closed range, the same is true for its scalar part. In this paper we answer this in the affirmative and point out some consequences concerning the essential spectra of prespectral operators. Also, following Taylor and Halberg [11], we present the state diagram of a prespectral operator, which will show, in a sense, the sharpness of the results of the spectral theory of such operators.

We describe two congruences α and γ contained in ℒ on an arbitrary orthodox semigroup. Let S be a right unipotent semigroup. We show that (i) α is an inverse semigroup congruence and γ is the finest fundamental inverse semigroup congruence on S, (ii) S is a union of groups if and only if ℒ on S and (iii) S is a band of groups if and only if ℒ on S.

In this paper we obtain the general solution of the equation

whose kernel is

where 0<α, β<1, μ, η, σ > 0, δ>0, are real parameters, z = max(x, y), φ(t) and g(y) are prescribed functions and ƒ(x) is to be determined.

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

In [8] the author studied the question of the primitivity of an Ore extension R[x, δ], where δ is a derivation of the ring R. If a is an automorphism of R then it can be shown that R[x, α] is primitive if the following conditions are satisfied: (i) no power αsS ≥ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are also known to be necessary and sufficient for the skew Laurent polynomial ring R[x, x−1, α] to be simple [9]. The object of this paper is to find conditions which are sufficient for R[x, x−1, α] to be primitive. The results obtained are remarkably similar to those of [8]. Two logically independent conditions are each found to be sufficient for the primitivity of R[x, x−1, α]. Of these, one is also shown to be sufficient for R[x, α] to be primitive. Included in the examples illustrating these results are some applications to the theory of primitive group rings. The basic techniques involved are also applied to produce a counterexample to the converse of a theorem of Goldie and Michler [3] on when R[x, x−1, α] is a Jacobson ring.

In [1] J. Lindenstrauss and M. A. Perles studied the extreme points of the set of all linear operators T of norm ≤ 1 from a finite dimensional Banach space X into itself. In particular they studied the question “When do these extreme points form a semigroup?”.

Interest in the ranges of Lyapunov transformations began with Taussky in [5]. Recently in a series of papers, Loewy has studied the ranges of Lyapunov transformations on matrices. In particular in [2] and [3], the following result was obtained.