C- and C^*-quotients in pointfree topology

Contracting the Socle in Rings of Continuous Functions

“…It will be recalled that L is the collection of frame homomorphisms from v(R) into L. See also [1] for some properties of the ring…”

“…Recall [1] that an onto frame homomorphism h: L 3 M is called a Cquotient map precisely when the ring homomorphism h is onto. Also, h is said to be coz-onto if for any c P Coz …”

Real Ideals in Pointfree Rings of Continuous Functions

“…To see maximality, let τ / ∈ M. Then coz τ / ∈ J , and so there exists c ∈ J such that coz τ ∨ c = 1. If γ ∈ RL is such that coz γ = c, then coz(τ 2 + γ 2 ) = 1, which implies that the ideal generated by τ and M is the entire ring. Now note that a n ∈ coz[M], for each n. Furthermore, ϕ / ∈ M, otherwise we have coz ϕ ∈ coz[M], and hence 1 = coz ϕ ∨ ϕ(−1, 1) ∈ coz[M], contrary to the fact that M is proper.…”

“…Therefore h( p) ∈ Pt(M). Pointfree function rings can be studied starting with OR, as in [2], or starting with the frame of reals L(R), as in [3]. We follow the latter approach.…”

A note on weakly pseudocompact locales

“…Since points of a sublocale are precisely the points of the locale belonging to the sublocale, there is a point p ∈ Pt(L) such that p ∈ G. Consequently, p ∈ o(a n ) for each n, so that c(p) = {p, 1} ⊆ o(a n ) and hence p ∨ a n = 1. Since compact regular locales are normal, for each n there exists c n ∈ Coz L such that c n ≤ p and c n ∨ a n = 1 (see [1,Corollary 8.3.2]). Put c = c n , so that c ∈ Coz L, c ≤ p < 1, and T. Dube c ∨ a n = 1 for every n. Consequently, c(c) ⊆ o(a n ) for every n, and so…”