In the
mathematical field of
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an element ''a'' of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
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 among the non-zero elements, or alternatively an element that
covers the least element 0.
Atomic orderings
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'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite partially ordered set with 0 is atomic, but the set of nonnegative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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'' < ''b'' there is an element ''c'' such that ''a'' <: ''c'' ≤ ''b'' or, equivalently, if every interval
'a'', ''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
least upper bound of a set of atoms. The linear order with three elements is not atomistic (see Fig. 2).
Atoms in partially ordered sets are abstract generalizations of
singletons in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
(see Fig. 1). Atomicity (the property of being atomic) provides an abstract generalization in the context of
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
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'' < 1 has a coatom ''c'' above it, and
* the set is coatomistic if every element is the
greatest lower bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of a set of coatoms.
References
*
External links
*
* {{planetmath reference, urlname=Poset, title=Poset
Order theory