## 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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