349,627 research outputs found

    Direct Product Decompositions of Lattices, Closures and Relation Schemes

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    In this paper we study direct product decompositions of closure operations and lattices of closed sets. We characterize direct product decompositions of lattices of closed sets in terms of closure operations, and find those decompositions of lattices which correspond to the decompositions of closures. If a closure on a finite set is represented by its implication base (i.e. a binary relation on a powerset), we construct a polynomial algorithm to find its direct product decompositions. The main characterization theorem is also applied to define direct product decompositions of relational database schemes and to find out what properties of relational databases and schemes are preserved under decompositions

    The Growth in Normal Subgroups Under Direct Products

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    This paper will consider the growth of the number of normal subgroups in a nonabelian group under the direct product operation. The groups considered in this paper are dihedral groups and semidirect products with a similar structure. Many proofs will rely on the use of a key property of normal subgroups, namely closure under conjugation, to predict this growth

    On the homotopy Lie algebra of an arrangement

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    Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincar\'e polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa

    Some combinatorial characteristics of closure operations

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    The aim of this paper investigates some combinatorial characteristics of minimal key and antikey of closure operations. We also give effective algorithms finding minimal keys and antikeys of closure operations. We estimate these algorithms. Some remarks on the closeness of closure operations class under the union and direct product operations are also studied in this paper

    On residuals of finite groups

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    A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in \cite{DHKL}, we establish the following generalisation: if X\mathfrak{X} is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X\overline{\mathfrak{X}} is the extension-closure of X\mathfrak{X}, then there exists an (optimal) constant γ\gamma depending only on X\mathfrak{X} such that, for all non-trivial finite groups GG with trivial X\mathfrak{X}-radical, \begin{equation} \left\lvert G^{\overline{\mathfrak{X}}}\right\rvert \,>\, \vert G\vert^\gamma, \end{equation} where GXG^{\overline{\mathfrak{X}}} is the X{\overline{\mathfrak{X}}}-residual of GG. When X=N\mathfrak{X} = \mathfrak{N}, the class of finite nilpotent groups, it follows that X=S\overline{\mathfrak{X}} = \mathfrak{S}, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.\,G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X\mathfrak{X} of full characteristic such that SXE\mathfrak{S} \subset \overline{\mathfrak{X}} \subset \mathfrak{E}, thus providing applications of our main result beyond the reach of \cite[Thm.~C]{DHKL}.Comment: 19 page
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