- FindStat

Identifier

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001615: Lattices ⟶ ℤ

Values

([],1) => ([],1) => 0

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

([(0,1)],2) => ([(0,1)],2) => 1

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

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

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

([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8) => 2

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8) => 2

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

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

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

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

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

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

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

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

([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8) => 0

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8) => 4

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

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

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

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

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

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

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

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

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

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8) => 2

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8) => 4

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

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8) => 4

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0

([(4,5)],6) => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8) => 0

([(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8) => 0

([(2,3),(2,4),(2,5)],6) => ([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8) => 0

([(1,2),(1,3),(1,4),(1,5)],6) => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8) => 1

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

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

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

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

([(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

([(1,3),(1,4),(1,5),(5,2)],6) => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2),(6,5)],8) => 1

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

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8) => 1

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(7,6)],8) => 2

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$F_{1} = 1$

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

$F_{3} = 1 + 4\ q^{2}$

$F_{4} = 4 + 2\ q + 5\ q^{2} + 4\ q^{3} + q^{4}$

$F_{5} = 13 + 16\ q + 15\ q^{2} + 10\ q^{3} + 9\ q^{4}$

Description

The number of join prime elements of a lattice.
An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Map

Dedekind-MacNeille completion

Description

Return the smallest lattice containing the poset.