- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 15:15:59
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=PM&vol=74&iss=3&update_since=2024-03-29
Portugaliae Mathematica
Port. Math.
PM
0032-5155
1662-2758
General
10.4171/PM
http://www.ems-ph.org/doi/10.4171/PM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2008)
74
2017
3
Special issue on computational algebra
João
Araújo
Universidade Aberta, Lisboa, Portugal and Universidade de Lisboa, Portugal
Peter
Cameron
University of St Andrews, UK
General
171
172
10.4171/PM/2000
http://www.ems-ph.org/doi/10.4171/PM/2000
2
8
2018
Two variants of the Froidure–Pin Algorithm for finite semigroups
Julius
Jonušas
Technische Universität Wien, Austria
James
Mitchell
University of St Andrews, UK
Markus
Pfeiffer
University of St Andrews, UK
Semigroups, monoids, algorithms, Green’s relations
In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If $U$ is any semigroup, and $A$ is a subset of $U$, then we denote by $\langle A\rangle$ the least subsemigroup of $U$ containing $A$. If $B$ is any other subset of $U$, then, roughly speaking, the first algorithm we present describes how to use any information about $\langle A\rangle$, that has been found using the Froidure-Pin Algorithm, to compute the semigroup $\langle A\cup B\rangle$. More precisely, we describe the data structure for a finite semigroup $S$ given by Froidure and Pin, and how to obtain such a data structure for $\langle A\cup B\rangle$ from that for $\langle A\rangle$. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.
Group theory and generalizations
173
200
10.4171/PM/2001
http://www.ems-ph.org/doi/10.4171/PM/2001
2
8
2018
Finding intermediate subgroups
Alexander
Hulpke
Colorado State University, Fort Collins, USA
Algorithm, interval, subgroups, lattice
This article describes a practical approach for determining the lattice of subgroups $U < V < G$ between given subgroups $U$ and $G$, provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.
Group theory and generalizations
Computer science
201
212
10.4171/PM/2002
http://www.ems-ph.org/doi/10.4171/PM/2002
2
8
2018
Synchronization and separation in the Johnson schemes
Mohammed
Aljohani
University of St Andrews, UK
John
Bamberg
University of Western Australia, Perth, Australia
Peter
Cameron
University of St Andrews, UK
Steiner system, association scheme, Johnson scheme, synchronization, separation, projective plane
Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary divisibility conditions on the parameters are satisfied and $n$ is sufficiently large in terms of $k$ and $t$. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash’s theorem, and to give some theoretical and computational evidence for the conjecture. We phrase the conjecture in terms of the notions (which we define here) of synchronization and separation for association schemes. These definitions are based on those for permutation groups which grow out of the theory of synchronization in finite automata. In this theory, two classes of permutation groups (called synchronizing and separating) lying between primitive and $2$-homogeneous are defined. A big open question is how the permutation group induced by $S_n$ on $k$-subsets of $\{1,\ldots,n\}$ fits in this hierarchy; our conjecture would give a solution to this problem for $n$ large in terms of $k$. We prove the conjecture in the case $k=4$: our result asserts that $S_n$ acting on $4$-sets is separating for $n\ge10$ (it fails to be synchronizing for $n=9$).
Group theory and generalizations
Combinatorics
213
232
10.4171/PM/2003
http://www.ems-ph.org/doi/10.4171/PM/2003
2
8
2018
On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups
Leonard
Soicher
Queen Mary University of London, UK
Classification of combinatorial objects, friendly subgroups, Sylow tower groups, partial spreads
We describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup $H$ of a group $K$ is a friendly subgroup of $K$ if every subgroup of $K$ isomorphic to $H$ is conjugate in $K$ to $H$. We explore easy-to-test sufficient conditions for a subgroup $H$ to be a friendly subgroup of a finite group $K$, and for this, present an algorithm for determining whether a finite group $H$ is a Sylow tower group. We also classify the maximal partial spreads invariant under a group of order $5$ in both PG(3,7) and PG (3,8).
Combinatorics
Group theory and generalizations
Geometry
233
242
10.4171/PM/2004
http://www.ems-ph.org/doi/10.4171/PM/2004
2
8
2018
Commutativity theorems for groups and semigroups
Francisco
Araújo
Colégio Planalto, Lisboa, Portugal
Michael
Kinyon
University of Denver, USA and Universidade de Lisboa, Portugal
Separative semigroup, inverse semigroup, completely regular, commutativity theorems
In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup $S$ we have $x^p y^p = y^p x^p$ and $x^q y^q = y^q x^q$ for all $x,y\in S$ where $p$ and $q$ are relatively prime, then $S$ is commutative. In a separative or inverse semigroup $S$, if there exist three consecutive integers $i$ such that $(xy)^i = x^i y^i$ for all $x,y\in S$, then $S$ is commutative. Finally, if $S$ is a separative or inverse semigroup satisfying $(xy)^3=x^3y^3$ for all $x,y\in S$, and if the cubing map $x\mapsto x^3$ is injective, then $S$ is commutative.
Group theory and generalizations
243
255
10.4171/PM/2005
http://www.ems-ph.org/doi/10.4171/PM/2005
2
8
2018
A new class of monohedral pentagonal spherical tilings with GeoGebra
Ana M.
d'Azevedo Breda
Universidade de Aveiro, Portugal
José
Dos Santos Dos Santos
Universidade Aberta, Lisboa, Portugal
Spherical geometry, spherical tilings, GeoGebra
By a monohedral spherical tiling we mean a decomposition of the sphere by geodesic congruent polygons. Here, making use of GeoGebra, a well known free interactive mathematics software, we show how to generate new classes of monohedral non-convex triangular and new non-convex pentagonal spherical tilings, changing the side gluing rules of the regular spherical tetrahedral tiling, by means of a local action of particular subgroups of spherical isometries. In both cases each face has $\pi$ as area measure. In relation to the new class of pentagonal tilings, we describe some of their properties and show the existence, in a special case, of an associated dihedral triangular spherical tiling, that is, a tiling composed by two sets of congruent triangles. These classes of spherical tilings have emerged as a result of an interative construction process, only possible by the use of newly produced GeoGebra tools and the dynamic interaction capabilities of this software.
Geometry
Global analysis, analysis on manifolds
257
266
10.4171/PM/2006
http://www.ems-ph.org/doi/10.4171/PM/2006
2
8
2018