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Quelle mgmadj.xml Sprache: XML |
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<ManSection> <Filt Name="IsMagmaWithZeroAdjoined" Arg='M' Type="Category"/> <Returns><K>true</K> or <K>false</K>.</Returns> <Description> <C>IsMagmaWithZeroAdjoined</C> returns <K>true</K> if the magma <A>M</A> was created using <Ref Attr="InjectionZeroMagma"/> or <Ref Attr="MagmaWithZeroAdjoined"/> and returns <K>false</K> if it was not. <Example><![CDATA[ gap> S:=Semigroup(Transformation([1,1,1]), Transformation([1,3,2]));; gap> IsMagmaWithZeroAdjoined(S); false gap> M:=MagmaWithZeroAdjoined(S); <<transformation semigroup of degree 3 with 2 generators> with 0 adjoined> gap> IsMagmaWithZeroAdjoined(M); true ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="InjectionZeroMagma" Arg='M'/> <Attr Name="MagmaWithZeroAdjoined" Arg='M'/> <Description> <Ref Attr="InjectionZeroMagma"/> returns an embedding from the magma <A>M</A> into a new magma formed from <A>M</A> by adjoining a single new element which is the multiplicative zero of the resulting magma. The elements of the new magma form a family of elements in the category <Ref Filt="IsMultiplicativeElementWithZero"/> and the magma itself satisfies <Ref Filt="IsMagmaWithZeroAdjoined"/>.<P/> <C>MagmaWithZeroAdjoined</C> is just shorthand for <C>Range(InjectionZeroMagma(<A>M</A>)))</C>.<P/> If <C>N</C> is a magma with zero adjoined, then the embedding used to create <C>N</C> can be recovered using <Ref Attr="UnderlyingInjectionZeroMagma"/>. <Example><![CDATA[ gap> S:=Monoid(Transformation( [ 7, 7, 5, 3, 1, 3, 7 ] ), > Transformation( [ 5, 1, 4, 1, 4, 4, 7 ] ));; gap> MultiplicativeZero(S); Transformation( [ 7, 7, 7, 7, 7, 7, 7 ] ) gap> T:=MagmaWithZeroAdjoined(S); <<transformation monoid of degree 7 with 2 generators> with 0 adjoined> gap> map:=UnderlyingInjectionZeroMagma(T);; gap> x:=Transformation( [ 7, 7, 7, 3, 7, 3, 7 ] );; gap> x^map; <monoid with 0 adjoined elt: Transformation( [ 7, 7, 7, 3, 7, 3, 7 ] )> gap> PreImage(map, x^map)=x; true ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="UnderlyingInjectionZeroMagma" Arg='M'/> <Description> <C>UnderlyingInjectionZeroMagma</C> returns the embedding used to create the magma with zero adjoined <A>M</A>. <Example><![CDATA[ gap> S:=Monoid(Transformation( [ 8, 7, 5, 3, 1, 3, 8, 8 ] ), > Transformation( [ 5, 1, 4, 1, 4, 4, 7, 8 ] ));; gap> MultiplicativeZero(S); Transformation( [ 8, 8, 8, 8, 8, 8, 8, 8 ] ) gap> T:=MagmaWithZeroAdjoined(S); <<transformation monoid of degree 8 with 2 generators> with 0 adjoined> gap> UnderlyingInjectionZeroMagma(T); MappingByFunction( <transformation monoid of degree 8 with 2 generators>, <<transformation monoid of degree 8 with 2 generators> with 0 adjoined>, function( elt ) ... end, function( x ) ... end ) ]]></Example> </Description> </ManSection> ¤ Dauer der Verarbeitung: 0.24 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28