Geodesic
Topological abelian groups with a Dedekind lattice of closed subgroups
Matematičeskie zametki, Tome 8 (1970) no. 4, pp. 509-519

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An investigation of the structure of locally compact abelian groups whose lattices of closed subgroups satisfy Dedekind's axiom. This condition is closely related to the finite-rank condition. In contrast to the discrete case, Dedekind groups form a relatively small subclass of the class of all locally compact abelian groups. A description is given of abelian groups such that the product of any two closed subgroups is closed. 
@article{MZM_1970_8_4_a10,
     author = {Yu. N. Mukhin},
     title = {Topological abelian groups with a {Dedekind} lattice of closed subgroups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {509--519},
     publisher = {mathdoc},
     volume = {8},
     number = {4},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_4_a10/}
}
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Yu. N. Mukhin. Topological abelian groups with a Dedekind lattice of closed subgroups. Matematičeskie zametki, Tome 8 (1970) no. 4, pp. 509-519. http://geodesic.mathdoc.fr/item/MZM_1970_8_4_a10/