scholarly journals Generating sets of an infinite semigroup of transformations preserving a zig-zag order

2020 ◽  
Vol 44 (6) ◽  
pp. 2132-2146
Author(s):  
Laddawan LOHAPAN ◽  
Jörg KOPPITZ ◽  
Somnuek WORAWISET
Keyword(s):  
Generating Sets ◽  
Semigroup Of Transformations

scholarly journals The rank of the semigroup of transformations stabilising a partition of a finite set

2015 ◽  
Vol 159 (2) ◽  
pp. 339-353 ◽  
Author(s):  
JOÃO ARAÚJO ◽  
WOLFRAM BENTZ ◽  
JAMES D. MITCHELL ◽  
CSABA SCHNEIDER
Keyword(s):  
Representation Theory ◽  
Upper Bound ◽  
Symmetric Groups ◽  
Complex Problem ◽  
Minimum Size ◽  
Generating Sets ◽  
Arbitrary Partition ◽  
Finite Set ◽  
Minimum Number ◽  
Semigroup Of Transformations

AbstractLet $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Main problems of import subsumption in domestic engineering, applicable to small-power electric generator installations

2018 ◽  
Vol 1 (1) ◽  
pp. 46-51 ◽  
Author(s):  
A. V. Shelgunov
Keyword(s):  
Low Power ◽  
Import Substitution ◽  
Power Generator ◽  
Small Power ◽  
Domestic Production ◽  
Power Generators ◽  
Electric Generator ◽  
Generating Sets ◽  
Current State ◽  
Generator Sets

Subject: the subject of the study are low-power generator sets with a power of up to 30 kW.Materials and methods: in this paper, the main domestic legislative documents regulating the requirements for products. An assessment is made of the current state of Russian engine building.Results: the detailed analysis of the modern domestic market of power generating units with a capacity of up to 30 kW is made, the main problems in the field of domestic production of  electric power generators in the range up to 30 kW are revealed, and the prospects for import substitution of gasoline and diesel engines are noted.Conclusions: almost complete absence of the market of domestic low-power generating sets is established, insufficient measures taken to support domestic producers are noted, measures are  proposed for the development of domestic production of power units in the range of up to 30 kW.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Sign in / Sign up

Export Citation Format

Share Document