7. The Complete Repertoire
We conclude by taking a very simple poset (P =[ZigZag[4]) and applying almost
every command to either P or J(P).
Build[ZigZag[4],zz4]
Build[JofP[zz4],jofzz4]
Building poset zz4 ...
Done
Building poset jofzz4 ...
Done
P[jofzz4]
PGraded[jofzz4]
{{}, {3}, {4}, {1, 3}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
{{{}}, {{3}, {4}}, {{1, 3}, {3, 4}}, {{1, 3, 4}, {2, 3, 4}},
{{1, 2, 3, 4}}}
RankedQ[jofzz4]
Rank[jofzz4]
NK[jofzz4]
RGF[jofzz4]
The poset is strongly ranked.
True
{0, 1, 1, 2, 2, 3, 3, 4}
{1, 2, 2, 2, 1}
2 3 4
1 + 2 q + 2 q + 2 q + q
Relabel[jofzz4,Compact]
Diagram[jofzz4,ShowLabels->True,
ThinLines->False,Jiggled->9]
-Graphics-
points = MaxAntichain[jofzz4]
links = DilworthCover[jofzz4]
Relabel[jofzz4]
Diagram[jofzz4,points,links,ShowLabels->True]
{6, 7}
{{1, 2}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {7, 8}}
-Graphics-
Fuse[DilworthCover[jofzz4]]
{{1, 2, 4, 6}, {3, 5, 7, 8}}
JoinSubLatticeQ[jofzz4,{2,4,5,7}]
MeetSubLatticeQ[jofzz4,{2,4,5,7}]
False
True
indices = JoinSubLattice[jofzz4,{2,4,5,7}]
Build[SubPoset[jofzz4,indices],jsub]
RestoreSubPosetLabels[jofzz4,indices,jsub]
Diagram[jsub,ShowLabels->True]
{2, 4, 5, 6, 7, 8}
Building poset jsub ...
Done
-Graphics-
Up[jofzz4]
Down[jofzz4]
{{1, 2, 3, 4, 5, 6, 7, 8}, {2, 4, 5, 6, 7, 8}, {3, 5, 6, 7, 8},
{4, 6, 8}, {5, 6, 7, 8}, {6, 8}, {7, 8}, {8}}
{{1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 2, 3, 5}, {1, 2, 3, 4, 5, 6},
{1, 2, 3, 5, 7}, {1, 2, 3, 4, 5, 6, 7, 8}}
CoverRelations[jofzz4]
Covers[jofzz4]
UpDegree[jofzz4]
CoCovers[jofzz4]
DownDegree[jofzz4]
{{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 5}, {4, 6}, {5, 6}, {5, 7},
{6, 8}, {7, 8}}
{{2, 3}, {4, 5}, {5}, {6}, {6, 7}, {8}, {8}, {}}
{2, 2, 1, 1, 2, 1, 1, 0}
{{}, {1}, {1}, {2}, {2, 3}, {4, 5}, {5}, {6, 7}}
{0, 1, 1, 1, 2, 2, 1, 2}
ZetaP[jofzz4]//MatrixForm
Mu[jofzz4]//MatrixForm
1 1 1 1 1 1 1 1
0 1 0 1 1 1 1 1
0 0 1 0 1 1 1 1
0 0 0 1 0 1 0 1
0 0 0 0 1 1 1 1
0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 1
1 -1 -1 0 1 0 0 0
0 1 0 -1 -1 1 0 0
0 0 1 0 -1 0 0 0
0 0 0 1 0 -1 0 0
0 0 0 0 1 -1 -1 1
0 0 0 0 0 1 0 -1
0 0 0 0 0 0 1 -1
0 0 0 0 0 0 0 1
ToRelation[ZetaP[jofzz4]]
{{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8},
{2, 2}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2, 8}, {3, 3}, {3, 5},
{3, 6}, {3, 7}, {3, 8}, {4, 4}, {4, 6}, {4, 8}, {5, 5}, {5, 6},
{5, 7}, {5, 8}, {6, 6}, {6, 8}, {7, 7}, {7, 8}, {8, 8}}
MaximalChainsUp[jofzz4]
MaximalChainsDown[jofzz4]
LinearExtensions[zz4]
{5, 3, 2, 1, 2, 1, 1, 1}
{1, 1, 1, 1, 2, 3, 2, 5}
{{2, 4, 1, 3}, {2, 1, 4, 3}, {2, 1, 3, 4}, {1, 2, 4, 3}, {1, 2, 3, 4}}
ChainsBetweenGF[jofzz4,1,Card[jofzz4]]
2 3 4
q + 6 q + 10 q + 5 q
ZetaPoly[jofzz4] // Together
Omega[zz4,n] // Together
2 3 4
2 n + 7 n + 10 n + 5 n
-------------------------
24
2 3 4
2 n + 7 n + 10 n + 5 n
-------------------------
24
OmegaGF[zz4,q]
OmegaBarGF[zz4,q]
2 3
q + 3 q + q
-------------
5
(1 - q)
2 3 4
q + 3 q + q
--------------
5
(1 - q)
G[zz4,q]
GBar[zz4,q]
2 3 4
1 + q + q + q + q
----------------------------------
2 3 4
(1 - q) (1 - q ) (1 - q ) (1 - q )
2 3 4 5 6
q + q + q + q + q
----------------------------------
2 3 4
(1 - q) (1 - q ) (1 - q ) (1 - q )
LatticeQ[jofzz4]
LJoin[jofzz4]//MatrixForm
LMeet[jofzz4]//MatrixForm
True
1 2 3 4 5 6 7 8
2 2 5 4 5 6 7 8
3 5 3 6 5 6 7 8
4 4 6 4 6 6 8 8
5 5 5 6 5 6 7 8
6 6 6 6 6 6 8 8
7 7 7 8 7 8 7 8
8 8 8 8 8 8 8 8
1 1 1 1 1 1 1 1
1 2 1 2 2 2 2 2
1 1 3 1 3 3 3 3
1 2 1 4 2 4 2 4
1 2 3 2 5 5 5 5
1 2 3 4 5 6 5 6
1 2 3 2 5 5 7 7
1 2 3 4 5 6 7 8
P[jofzz4]
Vee[jofzz4,3,4]
Wedge[jofzz4,3,4]
{1, 2, 3, 4, 5, 6, 7, 8}
6
1
Build[CP[jofzz4,Chain[4]],bigger]
Diagram[bigger]
Building poset bigger ...
Done
-Graphics-
Build[SubPoset[bigger,JI[bigger]],joinirreducibles]
Building poset joinirreducibles ...
Done
Diagram[joinirreducibles]
-Graphics-
Diagram[bigger,JI[bigger],JICoverRelations[bigger]]
-Graphics-
Diagram[bigger,MI[bigger],MICoverRelations[bigger]]
-Graphics-
ZetaMI[bigger]//MatrixForm
1 0 0 0 1 1 0
0 1 0 1 0 0 1
0 0 1 0 1 0 0
0 0 0 1 0 0 1
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
Build[Dual[jofzz4],jofzz4dual]
Diagram[jofzz4dual,ShowLabels->True]
RestoreDualPosetLabels[jofzz4,jofzz4dual]
Diagram[jofzz4dual,ShowLabels->True]
Building poset jofzz4dual ...
Done
-Graphics-
-Graphics-
IsomorphicQ[jofzz4,jofzz4dual]
The poset is strongly ranked.
Probably
Zap[zz4]
Zap[jofzz4]