Identifier
Values
=>
Cc0020;cc-rep-0 Cc0029;cc-rep
([(0,2),(1,2)],3)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(0,1),(0,2),(1,2)],3)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>2 ([(1,3),(2,3)],4)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(0,3),(1,2)],4)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>2 ([(2,4),(3,4)],5)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(1,4),(2,3)],5)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>2 ([(3,5),(4,5)],6)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(2,5),(3,4)],6)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>2 ([(4,6),(5,6)],7)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(3,6),(4,5)],7)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(4,5),(4,6),(5,6)],7)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>2
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Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Map
connected vertex partitions
Description
Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.