Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollák [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

In [10], C. Sundberg uses a clever argument involving an idea of Davie and Jewell [13] to prove an isomorphism theorem for a very general class of operators. A related spectral inclusion theorem is an immediate consequence of the proof of this result, as Sundberg points out. He goes on to list several well known examples that are applications of his main result and remarks that the proof of the McDonald–Sundberg theorem (c.f. [9]) can now be considerably simplified. The purpose of this note is to give further evidence of the utility of the criterion established in [10]. Here and throughout X denotes a compact Hausdorff space and A is a function algebra on X. The Shilovboundary of A is the minimal closed subset ∂(A) of X with the property that

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All representations and characters studied in this paper are taken over the complex numbers, and all groups considered are finite. For basic definitions concerning projective representations see [1].

If G is a group and or is a cocycle of G we denote by Proj(G, α) = {ξ1, …, ξt} the set of irreducible projective characters of G with cocycle α, where of course t is the number of α-regular conjugacy classes of G; ξ1, (1) is called the degree of ξ1. Also as normal, M(G) will denote the Schur multiplier of G, [α] the cohomology classof α, and [1] the cohomology class of the trivial cocycle of G.

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

Given a group presentation (or more generally† a 2-complex) one can associate with it an object which has variously been called the co-initial graph, star-graph, star-complex, and which has proved useful in several contexts [2], [6], [7], [8], [9], [10], [12]. For certain mappings of 2-complexes φ: ⃗ℒ (”strong mappings”) one gets an induced mapping φst: st⃗ℒst of the associated star-complexes. Then st is a covariant functor from the category of 2-complexes (where the morphisms are strong mappings) to the category of 1-complexes, and this functor behaves very nicely with respect to coverings (Theorem 1).

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

Let Gbe a finite group, and P:G → GL(n, ) be such that for all x, y ∈ G

(i) P(x)P(y) = α(x, y)P(xy), and

(ii)P(l) = In,

where α(x, y) ∈ *; then P is a projective representation of G with cocycle α and degree n. For other basic definitions concerning projective representations see [4].

In his paper [11], Peter Neumann considered in detail the cycle structures of elements of Aut(ℚ), the group of all homeomorphisms of the “rational world” ℚ onto itself, and further analyses of Aut(ℚ) and its subgroups have been given by Mekler [9], Bruyns [1], and Truss [13]. My interest in Aut(ℚ) stems from its utility in proving an at first sight rather startling (to a general topologist) result concerning β ℚ, the so-called Stone-Čech compactification of ℚ, namely that βℚ\ℚ is separable, and in fact contains a homogeneous countable dense subspace. (A space X is “homogeneous” provided whenever x, y ∈ X, there is some g ∈ Aut(X) with g(x) = y.) This is in sharp contrast to the spaces βℕ\ℕ and βℝ\ℝ, which are both inseparable.

Given a topological space X, we can consider the group G(X) of all autohomeomorphisms of X. Much is known about the relationship between X and G(X) for certain restricted classes of the space X; Whittaker [7] has shown that the existence of an isomorphism between any two sufficiently large subgroups of G(X) and G(Y) implies that X and Y are actually homeomorphic, whenever these are both compact, locally Euclidean manifolds, with or without boundary; Fine and Schweigert [1] give a detailed analysis of G(ℝ); recently, Neumann [4], Mekler [3] and Truss [6] have considered in depth the group G(ℚ).

Given a ring R and an injective ring endomorphism α: R → R, not necessarily surjective, it is possible to define a minimal overring A(R, α) of R to which extends as an automorphism. The ring A(R, α) was first studied by D. A. Jordan in his paper [5], where he also introduces the central objects of this paper—the closed left ideals of R. As can be seen from Theorem 4.7 of [5], the left ideal structure of A(R, α) depends very strongly on the closed left ideals of R, and our aim here is to show that each maximal left ideal of a left Artinian ring is closed.

The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X.

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Let u be the Thorn class of ξ in mod 2 cohomology. Then u is stably spherical by [2] and therefore all stable cohomology operations vanish on u, showing that wi(ξ)u = Sqiu = 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong mod p conditions for every prime p.

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

There are a number of theorems which bound d.l.(G), the derived length of a group G, in terms of the size of the set c.d.(G) of irreducible character degrees of G assuming that G is in some particular class of solvable groups ([1], [3], [4], [7]). For instance, Gluck [4] shows that d.l.(G)≤2 |c.d.(G)| for any solvable group, whereas Berger [1] shows that d.l.(G)≤|c.d.(G)| if G has odd order. One of the oldest (and smallest) such bounds is a theorem of Taketa [7] which says that d.l.(G)≤|c.d.(G)| if G is an M-group. Most of the existing theorems are an attempt to extend Taketa's bound to all solvable groups. However, it is not even known for M-groups whether or not this is the best possible bound. This suggests that given a class of solvable groups one might try to find the maximum derived length of a group with n character degrees (i.e. the best possible bound).

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

One of the fundamental results of representation theory is the identification of the irreducible representations of a semisimple group by their dominant weights [3]. The purpose of this paper is to establish similar results for a class of reductive algebraic monoids.

Let k be an algebraically closed field. An algebraic monoid is an affine algebraic variety M defined over k, together with an associative morphism m:M × M → M and a two-sided unit 1 ∈ M for m.

Let X and Y be normed spaces and let L(X, Y) denote the set of linear transformations (henceforth called “operators”) T with domain a linear subspace D(T) of X and range R(T) contained in Y. The restriction of T to a subspace E is denoted by T/E; by the usual convention T|E = T|E∩ D(T). For a given linear subspace E the family of infinite dimensional ssubspaces of E is denoted by (E). An operator Tis said to have a certain property ℙ ubiquitously if every E ∈ (X) contains an F ∈(E) for which T|F has property ℙ For example, T is ubiquitously continuous if each E ∈(X) contains an F∈ (E) for which T|F is continuous. In the present note we shall characterize ubiquitous continuity, isomorphy, precompactness and smallness. A subspace of X is called a principal subspace if it is closed and of finite codimension in X. The restriction of an operator to a principal subspace will be called a principal restriction. The symbol T will always denote an arbitrary operator in L(X, Y).