Nucleus (order theory)

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In mathematics, and especially in order theory, a nucleus is a function on a meet-semilattice such that (for every in ):[1]

Every nucleus is evidently a monotone function.

Frames and locales[edit]

Usually, the term nucleus is used in frames and locales theory (when the semilattice is a frame).

Proposition: If is a nucleus on a frame , then the poset of fixed points of , with order inherited from , is also a frame.[2]


  1. ^ Johnstone, Peter (1982), Stone Spaces, Cambridge University Press, p. 48, ISBN 978-0-521-33779-3, Zbl 0499.54001
  2. ^ Miraglia, Francisco (2006). An Introduction to Partially Ordered Structures and Sheaves. Polimetrica s.a.s. Theorem 13.2, p. 130. ISBN 9788876990359.