If Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal groups.

We investigate parabolic Higgs bundles and $\Gamma $-Higgs bundles on a smooth complex projective variety.

The paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

We give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

In this note a number of related problems about divisors are studied, and partial solutions obtained by elementary means. The problems are rather unconventional and seem to suggest interesting developments.

The coproduct of two frames A, B (equivalently the product of two locales, see Isbell (1972a)) will be shown to be given by the lattice of Galois connections between A and B (the connection product in the sense of Isbell (1972b)). We show the inverse limit of a direct system of locales to be given by the set inverse limit of the underlying lattices under the bonding antimaps (see Dowker and Strauss (1975)). This implies the existence of infinite locale products.

We introduce an approximation property (${\mathcal{K}}_{\mathit{up}}$-AP, $1\leq p<\infty$), which is weaker than the classical approximation property, and discover the duality relationship between the ${\mathcal{K}}_{\mathit{up}}$-AP and the ${\mathcal{K}}_{p}$-AP. More precisely, we prove that for every $1<p<\infty$, if the dual space $X^{\ast }$ of a Banach space $X$ has the ${\mathcal{K}}_{\mathit{up}}$-AP, then $X$ has the ${\mathcal{K}}_{p}$-AP, and if $X^{\ast }$ has the ${\mathcal{K}}_{p}$-AP, then $X$ has the ${\mathcal{K}}_{\mathit{up}}$-AP. As a consequence, it follows that every Banach space has the ${\mathcal{K}}_{u2}$-AP and that for every $1<p<\infty$, $p\neq 2$, there exists a separable reflexive Banach space failing to have the ${\mathcal{K}}_{\mathit{up}}$-AP.

G(n;l) will denote a graph of n vertices and l edges. Let f0(n, k) be the smallest integer such that there is a G (n;f0(n, k)) in which for every set of k vertices there is a vertex joined to each of these. Thus for example fo = 3 since in a triangle each pair of vertices is joined to a third. It can readily be checked that fo = 5 (the extremal graph consists of a complete 4-gon with one edge removed). In general we will prove: Let n > k, andthen f0(n, k) = f(n, k).

Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

A subset S of a ring R is a left semigroup ideal of R if RS ⊈ R, and a left ring ideal of R if in addition S is a subring of R. Gluskin (1960) investigated those rings with 1 which possess the property: (λ) every left semigroup ideal is a left ring ideal.

Let (G, +) be an arbitrary group and let T0(G) = {f∈ Map(G, G): 0f = 0}; the system composed of T0(G) and the operations of pointwise addition and composition of functions form a (left) near-ring. Berman and Silverman, in their investigation of near-rings of transformations [3], found that for every group G the associated near-ring of transformations T0(G) has no proper ideals. In the present paper left and right ideals of T0(G) are considered.

It is well known that a Boolean ring is isomorphic to a subdirect sum of two-element fields. In [3] a near-ring (B, +, ·) is said to be Boolean if there exists a Boolean ring (B, +, Λ, 1) with identity such that · is defined in terms of +, Λ, and 1 and, for any b ∈ B, b · b = b. A Boolean near-ring B is called special if a · b = (a ν x) Λ b, where x is a fixed element of B. It was pointed out that a special Boolean near-ring is a ring if and only if x = 0. Furthermore, a special Boolean near-ring does not have a right identity unless x = 0. It is natural to ask then whether any Boolean near-ring (which is not a ring) can have a right identity. Also, how are the subdirect structures of a special Boolean near-ring compared to those of a Boolean ring. It is the purpose of this paper to give a negative answer to the first question and to show that the subdirect structures of a special Boolean near- ring are very ‘close’ to those of a Boolean ring. In fact, we will investigate a class of near-rings that include the special Boolean near-rings and the Boolean semi- rings as defined in [8].

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

We investigate the relationship between the peripheral spectrum of a positive operator T on a Banach lattice E and the peripheral spectrum of the operators S dominated by T, that is, ]Sx] ≤ T]x] for all x ε E. This can be applied to obtain inheritance results for asymptotic properties of dominated operators.

We give some explicit constructions of type III product measures with various properties, resolving some conjectures of Brown, Dooley and Lake. We also define a family of Markov odometers of type III0 and show that the associated flow is approximately transitive.

A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.

It is assumed throughout this paper that all topological spaces under consideration are Hausdorff. Since the notion of topological property is fundamental in this paper, we begin by making it precise. For our purposes here, it is sufficient to think of a topological property Q as being a class of space such that if X ∈ Q and Y is homeomorphic to X, then Y∈Q. To say that a space X has property Q would then be equivalent to saying that X ∈ Q.

The object of this note is to study two properties of groups, which we will denote by (*) and (**). The property (*) is possessed by solvable groups (and in fact, by groups which have a solvable invariant system) and the property (**) is possessed by nilpotent groups (and in fact, by groups which have a central system).

Every lattice generated by three unordered elements contains a finite sublattice generated by three unordered elements. A list ℒ of twelve finite lattices, each generated by a three-element unordered set, is given. It is proved that every lattice generated by a three-element unordered set contains a sublattice isomorphic to onė of the lattices in ℒ moreover, ℒ is the smallest such list.