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For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...

(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.) I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...

(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...