Centered set

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. Note that any directed set is necessarily 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.

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", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 201, ISBN 978-0-521-78451-1, Zbl 1002.06001.