- FindStat

Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St000551: Lattices ⟶ ℤ

Values

['A',1] => ([],1) => ([],1) => 1
['A',2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 7
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 5

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$F_{1} = q$

$F_{2} = q^{4} + q^{5} + q^{7}$

Description

The number of left modular elements of a lattice.
A pair

$(x, y)$ of elements of a lattice

$L$ is a modular pair if for every

$z\geq y$ we have that

$(y\vee x) \wedge z = y \vee (x \wedge z)$ . An element

$x$ is left-modular if

$(x, y)$ is a modular pair for every

$y\in L$ .

Map

Dedekind-MacNeille completion

Description

Return the smallest lattice containing the poset.

Map

to root poset

Description

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where

$\alpha \prec \beta$ if

$\beta - \alpha$ is a simple root.