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Posets :: isAtomic

isAtomic -- determines if a lattice is atomic

Synopsis

Description

The lattice P is atomic if every non-minimal vertex of P is the join of atoms of P. Equivalently, P is atomic if every non-minimal, non-atom vertex of P covers at least two vertices.

The diamond poset is atomic. Also n booleanLattices are atomic.
i1 : P = poset {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}};
i2 : isLattice P

o2 = true
i3 : isAtomic P

o3 = true
i4 : isAtomic booleanLattice 4

o4 = true
The following lattice is non-atomic. Also, n chains are non-atomic, for n ≥3.
i5 : Q = poset {{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 6}, {5, 6}};
i6 : isLattice Q

o6 = true
i7 : isAtomic Q

o7 = false
i8 : isAtomic chain 5

o8 = false

See also

Ways to use isAtomic :