168 research outputs found
Standard automata and semidirect products of transformation semigroups
AbstractWe propose a decomposition of transformation semigroups (X, S) on a finite set X that provides 1.a composition of its elements out of idempotents/generators,2.a way by which S is obtained from semilattices/cyclic groups acting on X, namely by means of bilateral semidirect products and quotients.
The point is to provide both (a) and (b) simultaneously while still being accountable for the resources used in terms of cardinalities. This approach is applied to the semigroup End(X, ⩽) of isotonic mappings of a linearly ordered set as well as the transition semigroups of automata that arise from certain varieties of formal languages. We discuss the semigroup varieties D, R, J, LJ1, and give a bilateral semidirect decomposition of the full transformation semigroup T(X) into End (X, ⩽) and the symmetric group on X
On the isolated points in the space of groups
We investigate the isolated points in the space of finitely generated groups.
We give a workable characterization of isolated groups and study their
hereditary properties. Various examples of groups are shown to yield isolated
groups. We also discuss a connection between isolated groups and solvability of
the word problem.Comment: 30 pages, no figure. v2: minor changes, published version from March
200
Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
We prove that the noncrossing partition lattices associated with the complex
reflection groups for admit symmetric decompositions
into Boolean subposets. As a result, these lattices have the strong Sperner
property and their rank-generating polynomials are symmetric, unimodal, and
-nonnegative. We use computer computations to complete the proof that
every noncrossing partition lattice associated with a well-generated complex
reflection group is strongly Sperner, thus answering affirmatively a question
raised by D. Armstrong.Comment: 30 pages, 5 figures, 1 table. Final version. The results of the
initial version were extended to symmetric Boolean decompositions of
noncrossing partition lattice
Algebraic properties of profinite groups
Recently there has been a lot of research and progress in profinite groups.
We survey some of the new results and discuss open problems. A central theme is
decompositions of finite groups into bounded products of subsets of various
kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
On Fox and augmentation quotients of semidirect products
Let be a group which is the semidirect product of a normal subgroup
and some subgroup . Let , , denote the powers of the
augmentation ideal of the group ring . Using homological methods
the groups , , are
functorially expressed in terms of enveloping algebras of certain Lie rings
associated with and , in the following cases: for and arbitrary
(except from one direct summand of ), and for all if
certain filtration quotients of and are torsionfree.Comment: 39 pages; paper thoroughly revised: notation and presentation
improved, many details and new result added (Theorem 1.7
Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects
We consider three a priori totally different setups for Hopf algebras from
number theory, mathematical physics and algebraic topology. These are the Hopf
algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for
renormalization, and a Hopf algebra constructed by Baues to study double loop
spaces. We show that these examples can be successively unified by considering
simplicial objects, co-operads with multiplication and Feynman categories at
the ultimate level. These considerations open the door to new constructions and
reinterpretations of known constructions in a large common framework, which is
presented step-by-step with examples throughout. In this first part of two
papers, we concentrate on the simplicial and operadic aspects.Comment: This replacement is part I of the final version of the paper, which
has been split into two parts. The second part is available from the arXiv
under the title "Three Hopf algebras from number theory, physics & topology,
and their common background II: general categorical formulation"
arXiv:2001.0872
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