Lie algebras and separable morphisms in pro-affine algebraic groups

Let be an algebraically closed field of arbitrary characteristic, and let be a surjective morphism of connected pro-affine algebraic groups over . We show that if is bijective and separable, then is an isomorphism of pro-affine algebraic groups. Moreover, is separable if and only if (its differential) is surjective. Furthermore, if is separable, then .

     

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri].

We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions.

Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.     

Decidable (= separable) objects and morphisms in lextensive categories

After investigating all conceivable properties of decidable objects and maps in left exact categories with well-behaved finite sums (‘lextensive categories’), we give a characterization in such categories of decidable morphisms which are (finite) coverings (in an appropriate sense). Finally, we give two applications of this result, to separable algebras and to local homeomorphisms. In both cases it explains categorically the advantage of two well-known notions — strongly separable algebras and local homeomorphisms with path lifting property, respectively.     

Uniquely Clean Elements in Rings

It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Uniquely pairable graphs

The concept of a k-pairable graph was introduced by Z. Chen [On k-pairable graphs, Discrete Mathematics 287 (2004), 11-15] as an extension of hypercubes and graphs with an antipodal isomorphism. In the present paper we generalize further this concept of a k-pairable graph to the concept of a semi-pairable graph. We prove that a graph is semi-pairable if and only if its prime factor decomposition contains a semi-pairable prime factor or some repeated prime factors. We also introduce a special class of k-pairable graphs which are called uniquely k-pairable graphs. We show that a graph is uniquely pairable if and only if its prime factor decomposition has at least one pairable prime factor, each prime factor is either uniquely pairable or not semi-pairable, and all prime factors which are not semi-pairable are pairwise non-isomorphic. As a corollary we give a characterization of uniquely pairable Cartesian product graphs.     

<Emphasis Type="BoldItalic">κ</Emphasis>-Complete Uniquely Complemented Lattices

We show that for any infinite cardinal κ, every complete lattice where each element has at most one complement can be regularly embedded into a uniquely complemented κ-complete lattice. This regular embedding preserves all joins and meets, in particular it preserves the bounds of the original lattice. As a corollary, we obtain that every lattice where each element has at most one complement can be embedded into a uniquely complemented κ-complete lattice via an embedding that preserves the bounds of the original lattice.     

On separable self-complementary graphs

In this paper, we describe the structure of separable self-complementary graphs.     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

MacNeille completions and canonical extensions

Let be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety is generated by an elementary class of relational structures.

Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.

     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

An Explicit Example of a Reich Sequence for a Uniquely Extremal Quasiconformal Mapping

An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given.     

Permutations of separable preference orders

The notion of separability is important in economics, operations research, and political science, where it has recently been studied within the context of referendum elections. In a referendum election on n questions, a voter's preferences may be represented by a linear order on the 2ⁿ possible election outcomes. The symmetric group of degree 2ⁿ, S_2ⁿ, acts in a natural way on the set of all such linear orders. A permutation σ∈S_2ⁿ is said to preserve separability if for each separable order ?, σ(?) is also separable. Here, we show that the set of separability-preserving permutations is a subgroup of S_2ⁿ and, for 4 or more questions, is isomorphic to the Klein 4-group. Our results indicate that separable preferences are rare and highly sensitive to small changes. The techniques we use have applications to the problem of enumerating separable preference orders and to other broader combinatorial questions.     

Optimal objective function approximation for separable convex quadratic programming

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.Corresponding author.     

Transforming an error-tolerant separable matrix to an error-tolerant disjunct matrix

Recently, Chen and Hwang [H.B. Chen, F.K. Hwang, Exploring the missing link among d-separable, -separable and d-disjunct matrices, Discrete Applied Mathematics 133 (2007) 662-664] provided a method for transforming a separable matrix to a disjunct matrix. In [D.Z. Du, F.K. Hwang, Pooling Designs and Nonadaptive Group Testing — Important Tools for DNA Sequencing, World Scientific, 2006], Du and Hwang attempted to extend this result to its error-tolerant version; unfortunately, they gave an incorrect extension. This note gives a solution to this problem.     

Computational aspects of two-segment separable programming

Recursive separable programming algorithms based on local, two-segment approximations are described for the solution of separable convex programs. Details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach, and these approaches are compared. Computational comparisons of the methods are provided for a variety of test problems, including a water supply application (with more than 600 constraints and more than 900 variables) and an econometric modelling problem (with more than 200 variables). Research supported by National Science Foundation Grants MCS74-20584 A02 and MCS-7901066.     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy [15 Priddy, S. (1970). Koszul resolutions. Trans. Am. Math. Soc. 152:39–60.[Crossref] , [Google Scholar]]. There exist some relations between these algebras and the skew PBW extensions defined in [8 Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Comm. Algebra 39(1):50–75.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In [24 Suárez, H., Reyes, A. (submitted for publications). Koszulity for skew PBW extensions over fields. [Google Scholar]] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.