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Weak ideal topology in the topos of right acts over a monoid. (English) Zbl 1439.18004

Summary: Let \(S\) be a monoid. In this manuscript, our purpose is to study the notion of weak ideal topology \(j^I\) on the topos Act-\(S\) of all (right) representations of \(S\), where \(I\) is a left ideal of \(S\). After a brief analysis of the weak ideal topology, we give a necessary and sufficient condition for a \(j^I\)-separated \(S\)-act to become a \(j^I\)-sheaf in which the ideal \(I\) is central. Moreover, we establish another form of the double negation topology on Act-\(S\) which we call the torsion topology. Then, we retrieve the torsion topology on Act-\(S\) by means of the internal existential quantifier \(\exists_l:\mathrm{Rldl}(S)^l\to\mathrm{Rldl}(S)\), in which \(\mathrm{Rldl}(S)\) is the Heyting algebra of all right ideals of \(S\). Furthermore we give an explicit description of the associated sheaf functor for the ideal topology \(j^I\) where \(I\) is a central band of \(S\); e.g. the ideal \(\mathbb N\) of natural numbers of the monoid \((\mathbb N^\infty,\min)\) of extended natural numbers. Finally, for certain ideals \(I\) we show that the topos of all \(j^I\)-sheaves is a De Morgan topos provided that the monoid \(S\) satisfies in the right Ore condition.

MSC:

18B25 Topoi
06A15 Galois correspondences, closure operators (in relation to ordered sets)
20M12 Ideal theory for semigroups
20M30 Representation of semigroups; actions of semigroups on sets
20M50 Connections of semigroups with homological algebra and category theory
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