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**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 |

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\textit{M. L. H. Kilpack} and \textit{A. Magidin}, Commun. Algebra 49, No. 7, 2906--2915 (2021; Zbl 07409824)

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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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