Sums, products and negations of contexts and complete lattices

In Formal Concept Analysis, one associates with every context its concept lattice , and conversely, with any complete lattice L the standard context L, constituted by the join-irreducible elements as ‘objects’, the meet-irreducible elements as ‘attributes’, and the incidence relation induced by the lattice order. We investigate the effect of the operators and on various (finite or infinite) sum and product constructions. The rules obtained confirm the ‘exponential’ behavior of and the ‘logarithmic’ behavior of with respect to cardinal operations but show a ‘linear’ behavior on ordinal sums. We use these results in order to establish several forms of De Morgan’s law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context. Received February 7, 2001; accepted in final form January 6, 2006.     

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups*

Let be a class of groups. A subgroup H of a group G is called -s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ H_G belongs to where H_G is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to study some subgroups of Fitting subgroup and generalized Fitting subgroup -s-supplemented and some new criterions of p-nilpotency of finite groups are obtained. *This research is supported by the grant of NSFC and TianYuan Fund of Mathematics of China (Grant #10626047).     

Ideal-Triangularizability of Semigroups of Positive Operators

Let be a multiplicative semigroup of positive operators on a Banach lattice E such that every is ideal-triangularizable, i.e., there is a maximal chain of closed subspaces of E that consists of closed ideals invariant under S. We consider the question under which conditions the whole semigroup is simultaneously ideal-triangularizable. In particular, we extend a recent result of G. MacDonald and H. Radjavi. We also introduce a class of positive operators that contains all positive abstract integral operators when E is Dedekind complete.      

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

Lyndon’s groupoid generates a small almost Cross variety

Lyndon’s groupoid of order seven is the first published example of a non-finitely based finite algebra. The main objective of the present article is to investigate the variety generated by this groupoid and its subvarieties. It is shown that the subvarieties of form a chain of order five, all elements of which except are Cross varieties. It follows that the variety is also generated by a groupoid of order six and that any groupoid with five or fewer elements does not generate . Consequently, Lyndon’s example of a non-finitely based finite algebra could have been of order six instead of seven. It is also shown that, with respect to some important properties, Lyndon’s groupoid contrasts greatly with several well-known non-finitely based finite groupoids that were discovered shortly after its publication. Presented by R. Freese.     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

Peripherally Monomial-Preserving Maps between Uniform Algebras

Let and be uniform algebras and p(z,w) = z^mwⁿ a twovariable monomial. We characterize maps T from certain subsets of into such that holds for all f and g in the domain of T; peripherally monomial-preserving maps. Furthermore and are proved to be isometrical isomorphic as Banach algebras. If the greatest common divisor of m and n is 1, then T is extended to an isometrical linear isomorphism; a weighted composition operator. An example of peripherally monomial-preserving surjections between uniform algebras which is not linear, nor multiplicative, nor injective is given when the greatest common divisor is strictly greater than 1. The first, third and fourth authors were partly supported by the Grantsin-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.     

One-sided Grothendieck quotients

We introduce one-sided thick subcategories of an arbitrary preadditive category and define a quotient category . When is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick . We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory. Received: 14 November 2006 Revised: 12 March 2007     

Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

Given an action of a finite group G on a fusion category we give a criterion for the category of G-equivariant objects in to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain /2-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.      

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008     

On some vector lattices of operators and their finite elements

If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].      

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Every Biregular Function Is a Biholomorphic Map

We study Fueter-biregular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy–Riemann operator

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Value quotients and product torsion classes of lattice-ordered groups

For a set X of archimedean o-groups, the class of normal-valued ℓ-groups having the property that every value quotient is in X is a complete torsion class. A question of Conrad whether the class is a product torsion class is partially answered by investigating values and value quotients of products of normal-valued ℓ-groups. A major result is that the conjecture that all outer value quotients must be Dedekind complete is equivalent to the conjecture that measurable cardinals do not exist. Received January 20, 2007; accepted in final form February 22, 2008.     

On C *-Extreme Maps and *-Homomorphisms of a Commutative C *-Algebra

The generalized state space of a commutative C^*-algebra, denoted , is the set of positive unital maps from C(X) to the algebra of bounded linear operators on a Hilbert space . C^*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C^*-extreme point of satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C^*-extreme maps from C(X) into , the algebra generated by the compact and scalar operators, are multiplicative. This generalizes a result of D. Farenick and P. Morenz. We then determine the structure of these maps. This paper constitutes a part of the author’s Ph.D. thesis at the University of Nebraska-Lincoln.     

Genus formula for modular curves of {\mathcal{D}}-elliptic sheaves

We prove a genus formula for modular curves of -elliptic sheaves. We use this formula to show that the reductions of modular curves of -elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of tends to infinity. Received: 14 October 2008 The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

R-Boundedness of Smooth Operator-Valued Functions

In this paper we study R-boundedness of operator families , where X and Y are Banach spaces. Under cotype and type assumptions on X and Y we give sufficient conditions for R-boundedness. In the first part we show that certain integral operator are R-bounded. This will be used to obtain R-boundedness in the case that is the range of an operator-valued function which is in a certain Besov space . The results will be applied to obtain R-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Mark Veraar is supported by the Alexander von Humboldt foundation. His visit to Helsinki, which started this project, was funded by the Finnish Centre of Excellence in Analysis and Dynamics Research.