Let be a poset, partially ordered by . An element is called an atom if it covers some minimal element of . As a result, an atom is never minimal. A poset is called atomic if for every element that is not minimal has an atom such that .
Let be a set and its power set. is a poset ordered by with a unique minimal element . Thus, all singleton subsets of are atoms in .
is partially ordered if we define to mean that . Then is a minimal element and any prime number is an atom.
Remark. Given a lattice with underlying poset , an element is called an atom (of ) if it is an atom in . A lattice is a called an atomic lattice if its underlying poset is atomic. An atomistic lattice is an atomic lattice such that each element that is not minimal is a join of atoms. If is an atom in a semimodular lattice , and if is not under , then is an atom in any interval lattice where .
|Date of creation||2013-03-22 15:20:09|
|Last modified on||2013-03-22 15:20:09|
|Last modified by||CWoo (3771)|