In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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 int ...

, 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. A poset consists of a set together with a bina ...

with

least 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 dually, that is, it is an elem ...

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 mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically expres ...

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 ...

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 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 lo ...

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

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

s 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

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 dually, that is, it is an elem ...

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