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}};
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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 $n \geq 3$.
i5 : Q = poset {{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 6}, {5, 6}};
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i6 : isLattice Q o6 = true |
i7 : isAtomic Q o7 = false |
i8 : isAtomic chain 5 o8 = false |
The object isAtomic is a method function.