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atom (Definition)

Let $ P$ be a poset, partially ordered by $ \leq$. An element $ a\in P$ is called an atom if it covers some minimal element of $ P$. As a result, an atom is never minimal. A poset $ P$ is called atomic if for every element $ p\in P$ that is not minimal has an atom $ a$ such that $ a\leq p$.

Examples.

  1. Let $ A$ be a set and $ P=2^A$ its power set. $ P$ is a poset ordered by $ \subseteq$ with a unique minimal element $ \varnothing$. Thus, all singleton subsets $ \lbrace a \rbrace$ of $ A$ are atoms in $ P$.
  2. $ \mathbb{Z}^+$ is partially ordered if we define $ a\leq b$ to mean that $ a\mid b$. Then $ 1$ is a minimal element and any prime number $ p$ is an atom.

Remark. Given a lattice $ L$ with underlying poset $ P$, an element $ a\in L$ is called an atom (of $ L$) if it is an atom in $ P$. 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 $ a$ is an atom in a semimodular lattice $ L$, and if $ a$ is not under $ x$, then $ a\vee x$ is an atom in any interval lattice $ I$ where $ x=\bigwedge I$.

Examples.

  1. $ P=2^A$, with the usual intersection and union as the lattice operations meet and join, is atomistic: every subset $ B$ of $ A$ is the union of all the singleton subsets of $ B$.
  2. $ \mathbb{Z}^+$, partially ordered as above, with lattice binary operations defined by $ a\wedge b=\operatorname{gcd}(a,b)$, and $ a\vee b= \operatorname{lcm}(a,b)$, is a lattice that is atomic, as we have seen earlier. But it is not atomistic: $ 4$ is not a join of $ 2$'s; $ 36$ is not a join of $ 2$ and $ 3$ are just two counterexamples.



"atom" is owned by CWoo.
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Also defines:  atomic poset, atomic lattice, atomistic lattice, atomistic
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Cross-references: counterexamples, binary operations, meet, operations, union, intersection, interval, semimodular lattice, join, lattice, prime number, mean, subsets, singleton, power set, minimal, minimal element, covers, poset
There are 25 references to this entry.

This is version 10 of atom, born on 2005-06-14, modified 2006-08-23.
Object id is 7153, canonical name is Atom.
Accessed 6823 times total.

Classification:
AMS MSC: 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)
 06B99 (Order, lattices, ordered algebraic structures :: Lattices :: Miscellaneous)
Pending Errata and Addenda
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Point lattices by porton on 2006-08-22 12:47:14
It seems that the term "point lattice" is sometimes used in the same sense as "atomistic lattice" as defined here. Should put a note about this terminology.
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts
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