Centered set
Encyclopedia
In mathematics
, an upwards centered set A is a subset
of a partially ordered set
P, such that any finite subset of A has an upper bound
in P. Similarly, any finite subset of a downwards centered set has a lower bound. Note that any directed set
is neccessarily centered, and any centered set is linked
.
A subset B of a partial order is said to be σ-centered if it is a countable union
of centered sets.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an upwards centered set A is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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...
P, such that any finite subset of A has an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
in P. Similarly, any finite subset of a downwards centered set has a lower bound. Note that any directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
is neccessarily centered, and any centered set is linked
Linked set
In mathematics, an upwards linked set A is a subset of a partially ordered set P, in which any two of elements A have a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound. Note that every centered set is linked, which includes, in particular,...
.
A subset B of a partial order is said to be σ-centered if it is a countable union
Union
Union may refer to:* Trade union or labor union, an organization of workers that have banded together, often for the purpose of getting better working conditions or pay...
of centered sets.