Centered set
Order theory
In mathematics, in the area of order theory, an upwards centered set S is a subset of a partially ordered set, P, such that any finite subset of S has an upper bound in P. Similarly, any finite subset of a downwards centered set has a lower bound. An upwards centered set can also be called a consistent set. Any directed set is necessarily centered, and any centered set is a linked set.
A subset B of a partial order is said to be σ-centered if it is a countable union of centered sets.
References
- Fremlin, David H. (1984). Consequences of Martin's axiom. Cambridge tracts in mathematics, no. 84. Cambridge: Cambridge University Press. ISBN 0-521-25091-9.
- Davey, B. A.; Priestley, Hilary A. (2002), "9.1: Definitions", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 201, ISBN 978-0-521-78451-1, Zbl 1002.06001.
- v
- t
- e
Order theory
- Binary relation
- Boolean algebra
- Cyclic order
- Lattice
- Partial order
- Preorder
- Total order
- Weak ordering
- Antisymmetric
- Asymmetric
- Boolean algebra
- Completeness
- Connected
- Covering
- Dense
- Directed
- (Partial) Equivalence
- Foundational
- Heyting algebra
- Homogeneous
- Idempotent
- Lattice
- Reflexive
- Partial order
- Prefix order
- Preorder
- Semilattice
- Semiorder
- Symmetric
- Total
- Tolerance
- Transitive
- Well-founded
- Well-quasi-ordering (Better)
- (Pre) Well-order
- Alexandrov topology & Specialization preorder
- Ordered topological vector space
- Normal cone
- Order topology
- Order topology
- Topological vector lattice
- Antichain
- Cofinal
- Cofinality
- Comparability
- Duality
- Filter
- Hasse diagram
- Ideal
- Net
- Subnet
- Order morphism
- Order type
- Ordered field
- Ordered vector space
- Partially ordered group
- Upper set
- Young's lattice
![]() | This mathematical logic-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e