A curious nucleus
This is an interesting paper about the following curious feature of an arbitrary frame \(A\): Let \(\mathfrak{J}A\) denote the frame of ideals of \(A\) and consider the assignment \(\Xi:A\rightarrow \mathfrak{J}A\) given by \[ \Xi(a)=\{x\in A\mid (\forall y\in A)[x\vee y=\top\Rightarrow a\vee y=\top]\}. \] The operator \(\xi:A\rightarrow A\) defined by \(\xi(a)=\bigvee\Xi(a)\) is a pre-nucleus on \(A\). The paper contains a study of this pre-nucleus and the associated nucleus together with the important property of subfitness. When \(A\) is the frame \(\mathfrak{J}D\) of ideals of a (bounded) distributive lattice \(D\), \(\xi\) coincides with the nucleus \(j_C\) introduced by \textit{T. Coquand} in [``Compact spaces and distributive lattices'', J. Pure Appl. Algebra 184, No. 1, 1--6 (2003; Zbl 1034.54005)]. What does the pre-nucleus \(\xi\) do? It measures how near to being subfit the given frame \(A\) is: \(A\) is subfit if and only if \(\xi\) is as small as possible, that is, \(A\) is subfit if and only if \(\xi\) is the identity function on \(A\). For non-subfit frames a lower bound for \(\xi\) is given. Furthermore, for the case of a spatial frame \(A\), an upper bound for \(\xi\) is also given. The paper is very well written, easy to read, and poses various interesting open questions.
scientific article; zbMATH DE number 5726611
scientific article