De Morgan’s and strong De Morgan’s laws in a topos of sheaves. (English) Zbl 0644.18004

The logical principles \(\neg (x\wedge y)=\neg x\vee \neg y\) (de Morgan’s law) and \((x\to y)\vee (y\to x)=1\) (strong de Morgan’s law) do not hold in general for a locale L. These conditions were extensively investigated by P. T. Johnstone [Lect. Notes Math. 753, 479-491 (1979; Zbl 0445.03041)], where among other things, he established their connection with the notion of extremal disconnectedness in the topological (spatial) case. This short survey article lists various consequences that follow if a localic topos satisfies de Morgan’s or strong de Morgan’s laws. Proofs and details will appear in: “On primeness and maximality of filters” [Rapp., Sémin. Math. Pure, Univ. Cathol. Louvain 105 (1987)], as well as in some other forthcoming articles of the authors.
Reviewer: K.I.Rosenthal


18B25 Topoi
03G30 Categorical logic, topoi


Zbl 0445.03041
Full Text: EuDML