- FindStat

Identifier

Mp00193: Lattices —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001621: Lattices ⟶ ℤ

Values

([],1) => ([],1) => ([],1) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 1
([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

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$F_{1} = 1$

$F_{2} = q$

$F_{3} = q^{2}$

Description

The number of atoms of a lattice.
An element of a lattice is an atom if it covers the least element.

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.

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

$A$ .

Map

to poset

Description

Return the poset corresponding to the lattice.