349,627 research outputs found
Direct Product Decompositions of Lattices, Closures and Relation Schemes
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
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
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
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
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
is a subgroup-closed Fitting formation of full characteristic
which does not contain all finite groups and is the
extension-closure of , then there exists an (optimal) constant
depending only on such that, for all non-trivial finite
groups with trivial -radical, \begin{equation} \left\lvert
G^{\overline{\mathfrak{X}}}\right\rvert \,>\, \vert G\vert^\gamma,
\end{equation} where is the
-residual of . When , the class of finite nilpotent groups, it follows that
, 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 of full characteristic such that , thus providing
applications of our main result beyond the reach of \cite[Thm.~C]{DHKL}.Comment: 19 page
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