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.


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
Full Text: DOI


[1] Barr, M.; Wells, C., Toposes, triples and theories. Reprints in: Theory Appl. Categories, 12, 1-288, (2005) · Zbl 1081.18006
[2] Birkedal, L., Developing theories of types and computability via realizability, Electron. Notes Theoretical Comput. Sci., 34, 1-280, (2000) · Zbl 0947.68049
[3] Čech, E., Topologické Prostory (Prague 1959), English transl, (1966), Wiley, New York
[4] Ćirić, M., Lattices of subautomata and direct sum decompositions of automata, Algebra Colloq., 6, 1, 71-88, (1999) · Zbl 0943.68117
[5] Claes, V.; Sonck, G., The quasitopos hull of the construct of closure spaces, Appl. General Topol., 4, 1, 15-24, (2003) · Zbl 1048.18002
[6] Dikranjan, D.; Tholen, W., Categorical Structure of Closure Operators, (1995), Kluwer, Netherlands · Zbl 0853.18002
[7] Ebrahimi, M. M.; Mahmoudi, M., When is the category of separated M-sets a quasitopos or topos?, Bull. Iran. Math. Soc., 21, 25-33, (1995) · Zbl 0864.18002
[8] Ebrahimi, M. M., On ideal closure operators of M-sets, Southeast Asian Bull. Math., 30, 439-444, (2006) · Zbl 1111.18001
[9] Ehrig, H.; Herrlich, H., The construct PRO of projection spaces: its internal structure, Lect. Notes Comput. Sci., 393, 286-293, (1988)
[10] Español, L.; Lambán, L., On bornologies, locales and toposes of S-sets, J. Pure Appl. Algebra, 176, 113-125, (2002) · Zbl 1013.18002
[11] Giuli, E., On m-separated projection spaces, Appl. Categ. Struct., 2, 91-99, (1994) · Zbl 0840.18003
[12] Hosseini, S. N.; Mousavi, S. SH., A relation between closure operators on a small category and its category of presheaves, Appl. Categ. Struct., 14, 99-110, (2006) · Zbl 1102.18002
[13] Johnstone, P. T., Conditions Related to De Morgan’s Law, Applications of Sheaves. Lecture Notes in Math., Vol. 753, 479-491, (1979), Springer-Verlag, New York
[14] Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium, Vol. 1., (2002), Clarendon Press, Oxford · Zbl 1071.18001
[15] Kashu, A. I., On preradicals associated to principal functors of module categories. I, Bull. A. S. R. M. Math., 2, 60, 62-72, (2009) · Zbl 1201.16010
[16] Khanjanzadeh, Z.; Madanshekaf, A.
[17] Kilp, M.; Knauer, U.; Mikhalev, A. V., Monoids, Acts and Categories, (2000), Walter de Gruyter, Berlin · Zbl 0945.20036
[18] Komarnitskiy, M.; Oliynyk, R., Preradical and kernel functors over categories of S-acts, Algebra Discrete Math., 10, 1, 57-66, (2010) · Zbl 1212.20100
[19] Mac Lane, S.; Moerdijk, I., Sheaves in Geometry and Logic, (1992), Springer-Verlag, New York · Zbl 0822.18001
[20] Mahmoudi, M.; Moghaddasi Angizan, Gh., Sequentially injective hull of acts over idempotent semigroups, Semigroup Forum, 74, 240-246, (2007) · Zbl 1122.20033
[21] Zhang, R. Z.; Shum, K. P., Hereditary torsion classes of S-systems, Semigroup Forum, 52, 253-270, (1996) · Zbl 0851.20061
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