- FindStat

Identifier

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001719: Lattices ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,1),(6,2),(7,3),(7,4)],9) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1

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Description

The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.

Map

Dedekind-MacNeille completion

Description

Return the smallest lattice containing the poset.

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.