| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/openmath/tst/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 25.1.2023 mit Größe 15 kB |
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Quelle hasse.tst Sprache: unbekannt |
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gap> START_TEST("omput.tst");
######################################################## ## May also call `DrawHasse(diagram)` to produce html ######################################################## ## Bruhat order on permutations ######################################################## # return {(i,j) | i < j and p(i) > p(j)} gap> inv := function(p) > local n, i, j, res; > > res := []; > n := Length(ListPerm(p)); > > for j in [1 .. n] do > for i in [1 .. j-1] do > if i^p > j^p then > Append(res, [[i,j]]); > fi; > od; > od; > return res; > end; function( p ) ... end gap> gap> le := function(p,t) > return IsSubset(inv(t), inv(p)); > end; function( p, t ) ... end gap> gap> s := SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> l := AsSet(s); [ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3) ] gap> h := CreateHasseDiagram(l,le);; gap> OMPrint(h); <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> <OMBIND> <OMS cd="fns2" name="constant"/> <OMBVAR> <OMV name="()"/> <OMV name="(3,4)"/> <OMV name="(2,3)"/> <OMV name="(2,3,4)"/> <OMV name="(2,4,3)"/> <OMV name="(2,4)"/> <OMV name="(1,2)"/> <OMV name="(1,2)(3,4)"/> <OMV name="(1,2,3)"/> <OMV name="(1,2,3,4)"/> <OMV name="(1,2,4,3)"/> <OMV name="(1,2,4)"/> <OMV name="(1,3,2)"/> <OMV name="(1,3,4,2)"/> <OMV name="(1,3)"/> <OMV name="(1,3,4)"/> <OMV name="(1,3)(2,4)"/> <OMV name="(1,3,2,4)"/> <OMV name="(1,4,3,2)"/> <OMV name="(1,4,2)"/> <OMV name="(1,4,3)"/> <OMV name="(1,4)"/> <OMV name="(1,4,2,3)"/> <OMV name="(1,4)(2,3)"/> </OMBVAR> <OMA> <OMS cd="relation2" name="hasse_diagram"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="()"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(3,4)"/> <OMV name="(2,3)"/> <OMV name="(1,2)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(3,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,3,4)"/> <OMV name="(1,2)(3,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,4,3)"/> <OMV name="(1,2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,3,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,4)"/> <OMV name="(1,2,3,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,4,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,4)"/> <OMV name="(1,2,4,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(2,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2)(3,4)"/> <OMV name="(1,3,2)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2)(3,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,4,2)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,4,3)"/> <OMV name="(1,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,3,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,4)"/> <OMV name="(1,3,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,4,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3)(2,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,2,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,2,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,2)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3)"/> <OMV name="(1,4,3,2)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,4,2)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,4)"/> <OMV name="(1,4,2)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3)(2,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,2,4)"/> <OMV name="(1,4,2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,3,2,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)(2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,3,2)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,2)"/> <OMV name="(1,4,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,2)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)(2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4,2,3)"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)(2,3)"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="(1,4)(2,3)"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> gap> ######################################################## ## Divisor lattice of a natural number ######################################################## # try these numbers N := 102, 2618, 282387; # the number lattice is basically boring, just a concatenation of cubes... gap> leq := function(x,y) return y mod x = 0; end; function( x, y ) ... end gap> N := 102; 102 gap> h := CreateHasseDiagram(Filtered([1..N],i->N mod i = 0), leq);; gap> OMPrint(h); <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> <OMBIND> <OMS cd="fns2" name="constant"/> <OMBVAR> <OMV name="1"/> <OMV name="2"/> <OMV name="3"/> <OMV name="6"/> <OMV name="17"/> <OMV name="34"/> <OMV name="51"/> <OMV name="102"/> </OMBVAR> <OMA> <OMS cd="relation2" name="hasse_diagram"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="1"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="2"/> <OMV name="3"/> <OMV name="17"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="2"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="6"/> <OMV name="34"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="3"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="6"/> <OMV name="51"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="6"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="102"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="17"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="34"/> <OMV name="51"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="34"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="102"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="51"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="102"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="102"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> gap> ######################################################################### ## Some semigroups examples ############################################# ######################################################################### gap> s1 := Transformation([1,1,3,4]); Transformation( [ 1, 1 ] ) gap> s2 := Transformation([1,2,2,4]); Transformation( [ 1, 2, 2 ] ) gap> s3 := Transformation([1,2,3,3]); Transformation( [ 1, 2, 3, 3 ] ) gap> t1 := Transformation([2,2,3,4]); Transformation( [ 2, 2 ] ) gap> t2 := Transformation([1,3,3,4]); Transformation( [ 1, 3, 3 ] ) gap> t3 := Transformation([1,2,4,4]); Transformation( [ 1, 2, 4, 4 ] ) gap> o4 := Semigroup([s1,s2,s3,t1,t2,t3]); <transformation semigroup of degree 4 with 6 generators> gap> rcl := GreensRClasses( o4 ); [ <Green's R-class: Transformation( [ 1, 1, 1, 1 ] )>, <Green's R-class: Transformation( [ 1, 1, 1, 2 ] )>, <Green's R-class: Transformation( [ 1, 1, 2, 2 ] )>, <Green's R-class: Transformation( [ 1, 1, 2, 3 ] )>, <Green's R-class: Transformation( [ 1, 2, 2, 2 ] )>, <Green's R-class: Transformation( [ 1, 2, 2, 3 ] )>, <Green's R-class: Transformation( [ 1, 2, 3, 3 ] )> ] gap> h := CreateHasseDiagram(rcl, IsGreensLessThanOrEqual);; gap> OMPrint(h); <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> <OMBIND> <OMS cd="fns2" name="constant"/> <OMBVAR> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> </OMBVAR> <OMA> <OMS cd="relation2" name="hasse_diagram"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMV name="<object>"/> <OMV name="<object>"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMV name="<object>"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMV name="<object>"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMV name="<object>"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="<object>"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> gap> ################################################################### ## Some basic Hasse Diagrams - for when everything else is broken ################################################################### gap> d := Domain(["0","a","b","c","1"]);; gap> Elements(d); [ "0", "1", "a", "b", "c" ] gap> Size(d); 5 gap> r := BinaryRelationByElements(d, > [ DirectProductElement(["0","a"]), > DirectProductElement(["0","b"]), > DirectProductElement(["0","c"]), > DirectProductElement(["a","1"]), > DirectProductElement(["b","1"]), > DirectProductElement(["c","1"])]);; gap> SetIsHasseDiagram(r, true); gap> OMPrint(r); <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> <OMBIND> <OMS cd="fns2" name="constant"/> <OMBVAR> <OMV name="0"/> <OMV name="a"/> <OMV name="b"/> <OMV name="c"/> <OMV name="1"/> </OMBVAR> <OMA> <OMS cd="relation2" name="hasse_diagram"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="0"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="a"/> <OMV name="b"/> <OMV name="c"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="a"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="1"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="b"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="1"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="c"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="1"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="1"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> gap> ######################################################## ## Partitions of a Natural Number ## ## problematic? - writing matrices? ######################################################## # this identifies the holes where the spacers go gap> sumrep := function(p) > return List([1 .. Length(p)], i->Sum([1 .. i], j->p[j])); > end; function( p ) ... end # a refines b iff sumrep(b) is a subset of sumrep(a) gap> refines := function(a,b) > return IsSubset(sumrep(a), sumrep(b)); > end; function( a, b ) ... end gap> gap> p := Partitions(5); [ [ 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 2, 1 ], [ 3, 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5 ] ] gap> h := CreateHasseDiagram(p, refines);; gap> OMPrint(h); <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> <OMBIND> <OMS cd="fns2" name="constant"/> <OMBVAR> <OMV name="[ 1, 1, 1, 1, 1 ]"/> <OMV name="[ 2, 1, 1, 1 ]"/> <OMV name="[ 2, 2, 1 ]"/> <OMV name="[ 3, 1, 1 ]"/> <OMV name="[ 3, 2 ]"/> <OMV name="[ 4, 1 ]"/> <OMV name="[ 5 ]"/> </OMBVAR> <OMA> <OMS cd="relation2" name="hasse_diagram"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 1, 1, 1, 1, 1 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 2, 1, 1, 1 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 2, 1, 1, 1 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 2, 2, 1 ]"/> <OMV name="[ 3, 1, 1 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 2, 2, 1 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 4, 1 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 3, 1, 1 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 3, 2 ]"/> <OMV name="[ 4, 1 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 3, 2 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 5 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 4, 1 ]"/> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 5 ]"/> </OMA> </OMA> <OMA> <OMS cd="list1" name="list"/> <OMV name="[ 5 ]"/> <OMA> <OMS cd="list1" name="list"/> </OMA> </OMA> </OMA> </OMBIND> </OMOBJ> # gap> STOP_TEST("omput.tst"); [ Dauer der Verarbeitung: 0.39 Sekunden (vorverarbeitet) ] |
2026-03-28