A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that

(i)|X1|<|X|, for each i in I, and

(ii)|I|<|X|.

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

The study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. On the one hand, this class of algebras simultaneously abstracts de Morgan algebras and Stone algebras while, on the other hand, it has relevance to propositional logics lacking both the paradoxes of material implication and the law of double negation. Subsequently, these algebras were baptized distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [13]. In an elegant paper [9], M. S. Goldberg extended this theory and, amongst other things, described the free algebras and the injective algebras in those subvarieties of the variety 0 of distributive Ockham algebras which are generated by a single finite subdirectly irreducible algebra. Recently, T. S. Blyth and J. C. Varlet [4] explicitly described the subdirectly irreducible algebras in a small subvariety MS of 0 while in [2] the order-topological results of Goldberg were applied to accomplish the same objective for a subvariety k1.1 of 0 which properly contains MS. The aim, here, is to describe explicitly the injective algebras in each of the subvarieties of the variety MS. The first step is to draw the Hasse diagram of the lattice AMS of subvarieties of MS. Next, the results of Goldberg are applied to describe the injectives in each of the join irreducible members of AMS. Finally, this information, in conjunction with universal algebraic results due to B. Davey and H. Werner [8], is applied to give an explicit description of the injectives in each of the join reducible members of AMS.

We prove that up to isomorphism $\langle a,b\,\vert\, ab=0\rangle$ is the unique principal Rees quotient of a free inverse semigroup that is not trivial or monogenic with zero, satisfying a nontrivial identity in signature with involution.

Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra ←p(F), with a transparent combinatorial structure, which controls the rational representation theory of GL2(F).

Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

The goal of this paper is to describe explicitly simple modules for Schur superalgebra S(2|2) over an algebraically closed field K of characteristic zero or positive characteristic p>2.

We apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

Let A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

A closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

We prove a generalisation of an observation of N. Iwahori concerning the coefficients of the extended Dynkin diagram of a complex simple Lie algebra. We relate the combinatorics of these coefficients to the orders of finite groups that act discontinuously on the Riemann sphere and to the Plücker formulae.

Given a complete hereditary cotorsion pair $(\mathcal{X}, \mathcal{Y})$, we introduce the concept of $(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integer n, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤ n over a left Noetherian ring R. Similarly, if R is a left coherent ring in which all flat left R-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤ n. These structures have their analogous in the category of chain complexes.

This note extends classical results on certain Galois groups attached to onedimensional algebraic groups. We prove that the fields arising from the division of a fixed set of rational points on the product of an elliptic curve by the multiplicative group are as “large” as possible.

This paper may be regarded as a continuation of the investigations begun in [2]; certain intrinsic lattice topologies are studied, especially the order and ideal topologies in Boolean algebras, bicompactly generated lattices, and other more general structures. The results of [1], [2], and [3] are shown to be closely related. It is proved that the ideal topology on any Boolean algebra has a closed subbase consisting of all sublattices, whereas the order topology on an atomic Boolean algebra has a closed subbase consisting of all sub-complete lattices. It is also shown that the order topology on an atomic Boolean algebra is autouniformizable (in the sense defined by Rema [3]) and, if the ground set is infinite, strictly coarser than the ideal topology. The conditions Cl and C3 on a lattice, introduced by Kent [1], are shown to be slightly stronger than the condition “ bicompactly generated ”, and in complete lattices, where these conditions are satisfied, the order topology is shown to be coarser than the ideal topology.

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.

For any group G, denote by φf(G respectively L(G)) the intersection of all maximal subgroups of finite index (respectively finite nonprime index) in G, with the usual provision that the subgroup concerned equals Gif no such maximals exist. The subgroup φf(G) was discussed in [1] in connection with a property v possessed by certain groups: a group G has v if and only if every nonnilpotent, normal subgroup of G has a finite, nonnilpotent G-image. It was shown there, for instance, that G/φf(G) has v for all groups G. The subgroup L(G), in the case where G is finite, was investigated at some length in [3], one of the main results being that L(G) is supersoluble. (A published proof of this result appears as Theorem 3 of [4]). The present paper is concerned with the role of L(G) in groups G having property v or a related property a, the definition of which is obtained by replacing “nonnilpotent” by “nonsupersoluble” in the definition of v. We also present a result (namely Theorem 4) which displays a close relationship between the subgroups L(G) and φf(G) in an arbitrary group G. Some of the results for finite groups in [3] are found to hold with rather weaker hypotheses and, in fact, remain true for groups with v or a. We recall that if a group has a it also has v ([2]Theorem 2) but not conversely. For example, G = (x, y: y-1xy = x2)has v but not a. It is a well-known result of Gaschütz ([8], 5.2.15) that, in a finite group G, if His a normal subgroup containing φ(G) such that H/φ(G) is nilpotent than His nilpotent. This remains true in the case where G is any group with v [1, Proposition 1]. Our first result is in a similar vein and is a generalization of Theorem 9 of [7] and Theorem 1.2.9 of [3], the latter of which states that, for a finite group G, if G/L(G) is supersoluble, then so is G.

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations

,

for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

Ramanujan's short third notebook contains 33 pages of unorganized material. On page 386 of the third notebook, in the pagination of the Tata Institute's publication [3], Ramanujan offers the following remarkable identity.

A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

For an inverse semigroup S, the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by (S) and called the partial automorphism monoid of S. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that (S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on (S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.