Quelle semimaxplus.xml Sprache: XML

#############################################################################
##
#W  semimaxplus.xml
#Y  Copyright (C) 2016-2022                              Stuart A. Burrell
##
##  Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="IsXMatrixSemigroup">
  <ManSection Label = "IsXMatrixSemigroup">
  <Heading>Matrix semigroup filters</Heading>
    <Filt Name = "IsMatrixOverSemiringSemigroup"      Arg = "obj" Type = "Category"/>
    <Filt Name = "IsBooleanMatSemigroup"              Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMatrixOverFiniteFieldSemigroup"   Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMaxPlusMatrixSemigroup"           Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMinPlusMatrixSemigroup"           Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMatrixSemigroup"          Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMaxPlusMatrixSemigroup"   Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMinPlusMatrixSemigroup"   Arg = "obj" Type = "Category"/>
    
    <Filt Name = "IsNTPMatrixSemigroup"               Arg = "obj" Type = "Category"/>
    <Filt Name = "IsIntegerMatrixSemigroup"           Arg = "obj" Type = "Category"/>
    <Returns><K>true</K> or <K>false</K>.</Returns>
    <Description>
      The above are the currently supported types of matrix semigroups. For
      monoids see Section <Ref Sect = "IsXMatrixMonoid"/>.

    </Description>
  </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsXMatrixMonoid">
  <ManSection Label = "IsXMatrixMonoid">
    <Heading>Matrix monoid filters</Heading>
    <Filt Name = "IsMatrixOverSemiringMonoid"      Arg = "obj" Type = "Category"/>
    <Filt Name = "IsBooleanMatMonoid"              Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMatrixOverFiniteFieldMonoid"   Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMaxPlusMatrixMonoid"           Arg = "obj" Type = "Category"/>
    <Filt Name = "IsMinPlusMatrixMonoid"           Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMatrixMonoid"          Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMaxPlusMatrixMonoid"   Arg = "obj" Type = "Category"/>
    <Filt Name = "IsTropicalMinPlusMatrixMonoid"   Arg = "obj" Type = "Category"/>
    
    <Filt Name = "IsNTPMatrixMonoid"               Arg = "obj" Type = "Category"/>
    <Filt Name = "IsIntegerMatrixMonoid"           Arg = "obj" Type = "Category"/>
    <Returns><K>true</K> or <K>false</K>.</Returns>
    <Description>
      The above are the currently supported types of matrix monoids.  They
      correspond to the matrix semigroup types in
      Section <Ref Sect = "IsXMatrixSemigroup"/>.
    </Description>
  </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsFinite">
  <ManSection>
    <Prop Name = "IsFinite" Arg = "S"/>
    <Returns><K>true</K> or <K>false</K>. </Returns>
    <Description>
      If <A>S</A> is a max-plus or min-plus matrix semigroup (i.e. belongs to
      the category <Ref Filt = "IsMaxPlusMatrixSemigroup"/>), then
      <C>IsFinite</C> returns <K>true</K> if <A>S</A> is finite and
      <K>false</K> otherwise. This method is based on
      <Cite Key="Gaubert1996aa"/> (max-plus) and <Cite Key="Simon1978aa"/>
      (min-plus). For min-plus matrix semigroups, this method is only
      valid if each min-plus matrix in the semigroup contains only non-negative
      integers. Note, this method is terminating and does not involve
      enumerating semigroups.
      <Example><![CDATA[
gap> IsFinite(Semigroup(Matrix(IsMaxPlusMatrix,
>                              [[0, -3],
>                               [-2, -10]])));
true
gap> IsFinite(Semigroup(Matrix(IsMaxPlusMatrix,
>                              [[1, -infinity, 2],
>                               [-2, 4, -infinity],
>                               [1, 0, 3]])));
false
gap> IsFinite(Semigroup(Matrix(IsMinPlusMatrix,
>                              [[infinity, 0],
>                               [5, 4]])));
false
gap> IsFinite(Semigroup(Matrix(IsMinPlusMatrix,
>                              [[1, 0],
>                               [0, infinity]])));
true
]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsTorsion">
  <ManSection>
    <Attr Name = "IsTorsion" Arg = "S"/>
    <Returns><K>true</K> or <K>false</K>.</Returns>
    <Description>
      If <A>S</A> is a max-plus matrix semigroup (i.e. belongs to the
      category <Ref Filt = "IsMaxPlusMatrixSemigroup"/>), then
      <C>IsTorsion</C> returns <K>true</K> if <A>S</A> is torsion and
      <K>false</K> otherwise. This method is based on
      <Cite Key="Gaubert1996aa"/> and draws on <Cite Key="Burrell2016aa"/>,
      <Cite Key="Bacelli1992aa"/> and <Cite Key="Kacie2009aa"/>.

      <Example><![CDATA[
gap> IsTorsion(Semigroup(Matrix(IsMaxPlusMatrix,
>                              [[0, -3],
>                               [-2, -10]])));
true
gap> IsTorsion(Semigroup(Matrix(IsMaxPlusMatrix,
>                               [[1, -infinity, 2],
>                                [-2, 4, -infinity],
>                                [1, 0, 3]])));
false
]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

<#GAPDoc Label="NormalizeSemigroup">
  <ManSection>
    <Oper Name = "NormalizeSemigroup" Arg = "S"/>
    <Returns>A semigroup.</Returns>
    <Description>
      This method applies to max-plus matrix semigroups (i.e. those belonging to
      the category <Ref Filt = "IsMaxPlusMatrixSemigroup"/>) that are
      finitely generated, such that the spectral radius of the matrix equal to
      the sum of the generators (with respect to the max-plus semiring) is zero.
      <C>NormalizeSemigroup</C> returns  a semigroup of matrices all with
      strictly non-positive entries. Note that the output is isomorphic to a
      min-plus matrix semigroup. This method is based on
      <Cite Key="Gaubert1996aa"/> and  <Cite Key="Burrell2016aa"/>.
      <Example><![CDATA[
gap> NormalizeSemigroup(Semigroup(Matrix(IsMaxPlusMatrix,
>                                        [[0, -3],
>                                         [-2, -10]])));
<commutative semigroup of 2x2 max-plus matrices with 1 generator>
]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

quality100%

¤ Dauer der Verarbeitung: 0.21 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.