In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 Set (mathematics), 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 mathema ...
(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
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually ...
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 of a set of coatoms.
References
*
External links
*
* {{planetmath reference, urlname=Poset, title=Poset
Order theory