Identifier
Values
[[[]]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[]],[]] => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[]]] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[]]] => ([(0,4),(1,4),(2,4),(4,3)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[],[[],[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[],[]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[]],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[]],[],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[],[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[[],[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
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Description
The number of simple modules with projective dimension at most 1.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.
Map
lattice of congruences
Description
The lattice of congruences of a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.
Map
maximal antichains
Description
The lattice of maximal antichains in a poset.
An antichain $A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of $A$.
The set of maximal antichains can be ordered by setting $A \leq B \Leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow}B$, where $\mathop{\downarrow}A$ is the order ideal generated by $A$.
An antichain $A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of $A$.
The set of maximal antichains can be ordered by setting $A \leq B \Leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow}B$, where $\mathop{\downarrow}A$ is the order ideal generated by $A$.
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