## The lattice of algebraic closure operators on an infinite subgroup lattice.(English)Zbl 07409824

Summary: We say a lattice $$L$$ is a subgroup lattice if there exists a group $$G$$ such that $$\mathrm{Sub}(G)\cong L,$$ where $$\mathrm{Sub}(G)$$ is the lattice of subgroups of $$G$$, ordered by inclusion. We prove that the lattice of algebraic closure operators which act on the subgroup lattice of an infinite group is itself a subgroup lattice if and only if the group is isomorphic to the Prüfer $$p$$-group.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 20D30 Series and lattices of subgroups

### Keywords:

closure operator; subgroup lattice
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### References:

 [1] Birkhoff, G., Lattice Theory (1979), American Mathematical Society [2] Gorbunov, V. A., Algebraic Theory of Quasivarieties (1998), Siberian School of Algebra and Logic [3] Kilpack, M. L. H.; Magidin, A., The lattice of closure operators on a subgroup lattice, Commun. Algebra, 46, 4, 1387-1396 (2018) · Zbl 06891554 [4] Kilpack, M. L. H. (2012). The lattice of closure operators and the algebraic lattice of algebraic closure operators. PhD thesis. State University of New York at Binghamton, ProQuest LLC, Ann Arbor, MI. [5] Weak pseudo-complements of closure operators, Algebra Univers., 36, 3, 405-412 (1996) · Zbl 0901.06003
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