Definition
- A lattice (in combinatorics) is a poset \(\mathcal L\) so that every two elements \(a,b\in \mathcal L\) have a least upper bound (the join, \(a \vee b\)) and a greatest lower bound (the meet, \(a\wedge b\)). We assume all lattices have a unique minimal element \(\hat 0 \in \mathcal L\), which is true for finite lattices.
- An atom in a lattice \(\mathcal L\) with a unique minimal element \(\hat 0 \in \mathcal L\) is an element \(a\in \mathcal L\) so that \(\hat 0 < x \leq a\) implies \(x = a\).
- An atomic lattice is a lattice \(\mathcal L\) such that for all \(x\in \mathcal L\setminus \{\hat 0\}\) there exists an atom \(a\) with \(a\leq x\).
- An interval in a lattice is a subset
\[[a,b] = \{x\in \mathcal L ~ \mid ~ a \leq x \leq b\}.\]