A curious nucleus

Abstract

a b s t r a c t I describe a simple construction that assigns a pre-nucleus and associated nucleus to each frame. The nature of this nucleus appears to measure the subfitness properties of the parent frame. I pose a few questions concerning this pair.

More on a curious nucleus

“…It follows that F p ≤ ξ. We now show that it is in fact an equality, answering question (5) of Simmons [2010].…”

“…in Picado and Pultr [2012] and in Banaschewski [1988], where it is introduced) the term pre-nucleus is used for an inflator that satisfies the weaker property f (a) ∧ b ≤ f (a ∧ b). We follow Simmons [2010] in our choice of terminology.…”

“…In section 3 we study ξ on the frame ID of ideals of a distributive lattice D where ξ is idempotent and hence a nucleus. (This is the setting in Coquand [2003] which motivated the definition of ξ in Simmons [2010]. However the nucleus ξ in this setting has already appeared in Johnstone [1984].…”

“…However the nucleus ξ in this setting has already appeared in Johnstone [1984]. This reference was apparently missed by both Coquand [2003] and Simmons [2010].) Actually ID is the frame of open subsets of the spectrum SpecD of D. So a nucleus picks out the subspace of prime ideals fixed by it.…”

“…Following Simmons [2010] we compare ξ to another nucleus χ : ID → ID defined by χ(I) = c∈I {a ∈ D : a c} (note that here is computed in D rather than ID). Actually preserves (binary) meets and joins and therefore determines a congruence ≡ on D. From this it follows that the subspace of SpecD determined by χ is homeomorphic to Spec(D/≡).…”