Search results for "abelian"
showing 10 items of 208 documents
Butterflies in a Semi-Abelian Context
2011
It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp, with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show th…
A class of finite groups having nilpotent injectors
1986
AbstractThe purpose of this paper is to construct a class of groups which properly contains the class of N-constrained groups, and which is such that all groups in this class have N-injectors.
Associative rings with metabelian adjoint group
2004
Abstract The set of all elements of an associative ring R, not necessarily with a unit element, forms a monoid under the circle operation r∘s=r+s+rs on R whose group of all invertible elements is called the adjoint group of R and denoted by R°. The ring R is radical if R=R°. It is proved that a radical ring R is Lie metabelian if and only if its adjoint group R° is metabelian. This yields a positive answer to a question raised by S. Jennings and repeated later by A. Krasil'nikov. Furthermore, for a ring R with unity whose multiplicative group R ∗ is metabelian, it is shown that R is Lie metabelian, provided that R is generated by R ∗ and R modulo its Jacobson radical is commutative and arti…
Partial Finitely Generated Bi-Ideals
2016
Partial words have been studied by Blanchet-Sadri et al., but bi-ideals or reccurrent words have been studied for centuries by many researchers. This paper gives a solution for some problems for partial reccurrent words. This paper gives an algorithm for a given finitely generated bi-ideal, how to construct a new basis of ultimately finitely generated bi-ideal, which generates the same given bi-ideal. The paper states that it is always possible to find a basis for a given finitely generated bi-ideal. The main results of this paper are presented in third section. At first, we show that if two irreduciable bi-ideals are different, they will differ in infinitely many places. This led to the st…
A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules
2014
In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.
On the exterior degree of the wreath product of finite abelian groups
2013
The exterior degree $d^\wedge(G)$ of a finite group $G$ has been recently introduced by Rezaei and Niroomand in order to study the probability that two given elements $x$ and $y$ of $G$ commute in the nonabelian exterior square $G \wedge G$. This notion is related with the probability $d(G)$ that two elements of $G$ commute in the usual sense. Motivated by a paper of Erovenko and Sury of 2008, we compute the exterior degree of a group which is the wreath product of two finite abelian $p$--groups ($p$ prime). We find some numerical inequalities and study mostly abelian $p$-groups.
Algorithms for Computing Abelian Periods of Words
2012
Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\sigma$ can have $\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \times \sigma)$ using $O(n \times \sigma)$ space. We present an off-line algorithm based on a $\sel$ function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present on-line algorithms that also enable to compute all the Abelian periods of all the prefixes of $w$.
A note on easy and efficient computation of full abelian periods of a word
2016
Constantinescu and Ilie (Bulletin of the EATCS 89, 167-170, 2006) introduced the idea of an Abelian period with head and tail of a finite word. An Abelian period is called full if both the head and the tail are empty. We present a simple and easy-to-implement $O(n\log\log n)$-time algorithm for computing all the full Abelian periods of a word of length $n$ over a constant-size alphabet. Experiments show that our algorithm significantly outperforms the $O(n)$ algorithm proposed by Kociumaka et al. (Proc. of STACS, 245-256, 2013) for the same problem.
Abelian combinatorics on words: A survey
2022
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.
Abelian Repetitions in Sturmian Words
2012
We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of period $F_j$, $j>1$, has length $F_j(F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j(F_{j+1}+F_{j-1}…