Divisibility theory and complexity of algorithms for free partially commutative groups

Esyp, E. S.; Kazachkov, Ilya; Remeslennikov, Vladimir N.

doi:10.1090/conm/378/07014

Cited by 36 publications

(90 citation statements)

References 48 publications

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“…[38, Sect. 6.5]); • by removing the braid relations (3), and keeping idempotence (1) and far commutativity (2), we obtain the locally free idempotent monoid [67] (see also [24]); • by introducing the generators' inverses g −1 t , and replacing the idempotence relations (1) with cancellation relations g t g −1 t = id, we obtain the classical braid group.…”

Section: Theoremmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n 2 ). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n 1.5 ). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(n log n), thus approaching an answer to Landau's question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(n log 2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

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Section: Theoremmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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“…Further information on this class of groups may be found in [4], [5] and [6]. We say G is a (free) partially commutative nilpotent group of class 2 if G = X|R where…”

Section: G = X|{[[x Y] Z] : X Y Z ∈ X}mentioning

confidence: 99%

“…This implies that there exists some g ∈ G such that (i) [g, x i j ] G = 1 for all 1 j m; and, (ii) [g, x i m+1 ] G = 1. In [6], it is shown that if H is a free partially commutative group and x, h ∈ H with x a generator, then we have …”

Section: Proofmentioning

confidence: 99%

Centraliser Dimension of Free Partially Commutative Nilpotent Groups of Class 2

Blatherwick

2008

Glasgow Math. J.

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Abstract.In an effort to extend the theory of algebraic geometry over groups beyond free groups, Duncan, Kazatchkov and Remeslennikov have studied the notion of centraliser dimension for free partially commutative groups. In this paper we consider the centraliser dimension of free partially commutative nilpotent groups of class 2, showing that a free partially commutative nilpotent group of class 2 with non-commutation graph has the same centraliser dimension as the free partially commutative group represented by the non-commutation graph .2000 Mathematics Subject Classification. Primary 20F18; 20F36. Introduction.There has been much interest in recent years in the development of algebraic geometry over groups (see [1, 10, 12]), inspired by various successes in the study of equations over groups, such as Makanin's algorithm for deciding whether a system of equations over a free group has a solution, and the solution of Tarski's problem concerning the elementary theory of free groups by Kharlampovich and Myasnikov [9], and, independently, Sela [16]. In [3], Chiswell and Remeslennikov showed that if V is an irreducible algebraic set defined by a one-variable system of equations over a non-abelian free group, F, then either V = F, or V is a point, or there exist elements f, g, h ∈ F such that V = f C F (g)h, where C F (g) is the centraliser in F of g. It is therefore believed that the investigation of the structure of the centraliser lattice of a group is a first step in the construction of algebraic geometry over that group. The centraliser dimension of a group is defined to be the height of its centraliser lattice, a concept which has been studied in numerous papers, for example [15], [11], [2], [13] and [4]. To extend the theory of algebraic geometry over groups beyond free groups, Duncan, Kazatchkov and Remeslennikov [5] have studied the notion of centraliser dimension for free partially commutative groups, also known as graph groups, trace groups or right-angle Artin groups.Given an arbitrary variety of groups, V, we may define the class of free partially commutative V groups in an obvious fashion. In this paper we consider the centraliser dimension of free partially commutative nilpotent groups of class 2. We begin, in Section 3, by considering the form of powers and roots of group elements. We then investigate the structure of centralisers in free partially commutative nilpotent groups of class 2 in Section 4. In Section 5 we show that to calculate centraliser dimension in these groups we only need to consider chains of centralisers of generating elements. This result is then used to show that a free partially commutative nilpotent group of class 2 with non-commutation graph has the same centraliser dimension as the free partially commutative group represented by non-commutation graph .

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“…First we recall some of the notation and definitions of [8]. If w ∈ (X ∪ X −1 ) * then we denote by α(w) the set of elements x ∈ X such that x or x −1 occurs in w. By abuse of notation we identify words of (X ∪ X −1 ) * with the elements of G which they represent.…”

Section: Preliminariesmentioning

confidence: 99%

“…We shall also need the following lemma, which is based on Proposition 5.7 of [8], and its corollary. Proof.…”

Section: By φ(Y ) the Full Subgraph Of φ With Vertex Set Y ; So φ(Y )mentioning

confidence: 99%