Atomic (order theory)
Encyclopedia
In the mathematical field of order theory
, given two elements a and b of a partially ordered set
, one says that b covers a, and writes a <: b or b :> a, if a < b and there is no element c such that a < c < b. In other words, b covers a if b is greater than a and minimal with this property, or equivalently if a is smaller than b and maximal with this property.
In a partially ordered set with least element 0, an atom is an element that covers 0, i.e. an element that is minimal among the non-zero elements. A partially ordered set with a least element is called atomic if every non-zero element b > 0 has an atom a below it, i.e. b ≥ a :> 0.
A partially ordered set is called 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.
A partially ordered set with least element 0 is called atomistic if every element is the least upper bound of a set of atoms. Every finite poset is relatively atomic, and every finite poset with 0 is atomic. But the linear order with three elements is not atomistic.
Atoms in partially ordered sets are abstract generalizations of singletons in set theory
. 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.
The terms coatom, coatomic, and coatomistic are defined dually; thus in a partially ordered set with greatest element
1:
Order theory
Order theory is a branch of mathematics which investigates our 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 introduces the field and gives some basic definitions...
, given two elements a and b of a partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set 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 binary relation that indicates that, for certain pairs of elements in the...
, one says that b covers a, and writes a <: b or b :> a, if a < b and there is no element c such that a < c < b. In other words, b covers a if b is greater than a and minimal with this property, or equivalently if a is smaller than b and maximal with this property.
In a partially ordered set with least element 0, an atom is an element that covers 0, i.e. an element that is minimal among the non-zero elements. A partially ordered set with a least element is called atomic if every non-zero element b > 0 has an atom a below it, i.e. b ≥ a :> 0.
A partially ordered set is called 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.
A partially ordered set with least element 0 is called atomistic if every element is the least upper bound of a set of atoms. Every finite poset is relatively atomic, and every finite poset with 0 is atomic. But the linear order with three elements is not atomistic.
Atoms in partially ordered sets are abstract generalizations of singletons in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. Atomicity (the property of being atomic) provides an abstract generalization in the context of order theory
Order theory
Order theory is a branch of mathematics which investigates our 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 introduces the field and gives some basic definitions...
of the ability to select an element from a non-empty set.
The terms coatom, coatomic, and coatomistic are defined dually; thus in a partially ordered set with greatest element
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
1:
- A coatom is an element covered by 1
- The set is called coatomic if every b < 1 has a coatom c above it
- The set is called coatomistic if every element is the greatest lower bound of a set of coatoms.