Editing Atom (order theory)

In the [[Mathematics|mathematical]] field of [[order theory]], an element ''a'' of a [[partially ordered set]] with [[least element]] '''0''' is an '''atom''' if '''0''' < ''a'' and there is no ''x'' such that '''0''' < ''x'' < ''a''. 

Equivalently, one may define an atom to be an element that is [[minimal element|minimal]] among the non-zero elements, or alternatively an element that [[covering relation|covers]] the least element '''0'''.

==Atomic orderings==
{| style="float:right"
| [[File:Lattice T 4.svg|thumb|500x150px|'''Fig.&nbsp;2''': The [[lattice (order)|lattice]] of divisors of 4, with the ordering "''is [[divisor]] of''", is atomic, with 2 being the only atom and coatom. It is not atomistic, since 4 cannot be obtained as [[least common multiple]] of atoms.]]
|}
{| style="float:right"
| [[File:Hasse diagram of powerset of 3.svg|thumb|x150px|'''Fig.&nbsp;1''': The [[power set]] of the set {''x'', ''y'', ''z''} with the ordering "''is [[subset]] of''" is an atomistic partially ordered set: each member set can be obtained as the [[union (set theory)|union]] of all [[Singleton (mathematics)|singleton]] sets below it.]]
|}
Let <: denote the [[covering relation]] in a partially ordered set.

A partially ordered set with a least element '''0''' is '''atomic''' if every element ''b''&nbsp;>&nbsp;'''0''' has an atom ''a'' below it, that is, there is some ''a'' such that ''b''&nbsp;≥&nbsp;''a''&nbsp;:>&nbsp;''0''.  Every finite partially ordered set with '''0''' is atomic, but the set of nonnegative [[real number]]s (ordered in the usual way) is not atomic (and in fact has no atoms).

A partially ordered set is '''relatively atomic''' (or ''strongly atomic'') if for all ''a''&nbsp;<&nbsp;''b'' there is an element ''c'' such that ''a''&nbsp;<:&nbsp;''c''&nbsp;≤&nbsp;''b'' or, equivalently, if every interval [''a'',&nbsp;''b''] is atomic. Every relatively atomic partially ordered set with a least element is atomic. Every finite poset is relatively atomic.

A partially ordered set with least element '''0''' is called '''atomistic''' (not to be confused with '''atomic''') if every element is the [[Infimum_and_supremum#Suprema|least upper bound]] of a set of atoms. The linear order with three elements is not atomistic (see Fig.&nbsp;2).

Atoms in partially ordered sets are abstract generalizations of [[Singleton (mathematics)|singleton]]s in [[set theory]] (see Fig.&nbsp;1). Atomicity (the property of being atomic) provides an abstract generalization in the context of [[order theory]] of the ability to select an element from a non-empty set.

==Coatoms==
The terms ''coatom'', ''coatomic'', and ''coatomistic'' are defined dually.  Thus, in a partially ordered set with [[greatest element]] '''1''', one says that
* a '''coatom''' is an element covered by '''1''',
* the set is '''coatomic''' if every ''b''&nbsp;<&nbsp;'''1''' has a coatom ''c'' above it, and
* the set is '''coatomistic''' if every element is the [[greatest lower bound]] of a set of coatoms.

==References==
* {{Citation | last1=Davey | first1=B. A. | last2=Priestley | first2=H. A. |author2link = Hilary Priestley| title=Introduction to Lattices and Order|title-link= Introduction to Lattices and Order | publisher=[[Cambridge University Press]] | isbn=978-0-521-78451-1 | year=2002}}

==External links==
* {{planetmath reference|urlname=Atom|title=Atom}}
* {{planetmath reference|urlname=Poset|title=Poset}}

[[Category:Order theory]]