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##
#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 = "IsProjectiveMaxPlusMatrixSemigroup" 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 = "IsProjectiveMaxPlusMatrixMonoid" 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"/>.
<#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>
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