The maximality of filters (Q1174505)

This beautifully written and interesting paper deals with constructive characterization of prime filters on a Boolean algebra and of prime filters on a compact regular locale, both in an arbitrary topos. As such characterizations are stated in terms of maximality of the filters in question, they could hold without the assumption that the topos is a De Morgan topos [cf. \textit{P. T. Johnstone}, Commun. Algebra 7, 1309-1312 (1979; Zbl 0417.18002)] only if the notion of a maximal filter is appropriately defined. This is, in fact, the key aspect of this paper. Explicitly, a (proper) filter \(P\) on a Boolean algebra \(B\) in a topos \({\mathcal E}\) is here said to be a maximal filter if it satisfies internally a condition to the effect that for any filter \(F\) on \(B\) containing \(P\) and any \(a\in B\), the statement \(a\in F\to(a\in P\lor 0\in F)\) holds. It is shown that for any Boolean algebra \(B\) in the topos \({\mathcal E}\), the prime filters on \(B\) are exactly the maximal filters on \(B\) (in the above sense). In a similar vein, a result concerning the completely prime filters on a compact regular locale in a topos is shown. There is a connection between the proposed notion of maximality of filters and the De Morgan property of a topos: the proposed notion agrees with the usual if and only if the topos (in which both are interpreted) is a De Morgan topos.