Geodesic
The Löwenheim–Skolem–Mal'tsev Theorem for $\mathbb{HF}$-Structures
Algebra i logika, Tome 43 (2004) no. 6, pp. 749-758

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We deal with the problem asking whether hereditarily finite superstructures have elementary extensions of the form $\mathbb{HF}(\mathfrak M)$. In so doing, we settle the question whether a theory for some hereditarily finite superstructure have $\mathbb{HF}(\mathfrak M)$ models of arbitrarily large cardinality. A Hanf number is shown to exist, and we provide an exact bound for the countable case.
Keywords: hereditarily finite superstructure, Hanf number. 
@article{AL_2004_43_6_a5,
     author = {V. G. Puzarenko},
     title = {The {L\"owenheim{\textendash}Skolem{\textendash}Mal'tsev} {Theorem} for $\mathbb{HF}${-Structures}},
     journal = {Algebra i logika},
     pages = {749--758},
     publisher = {mathdoc},
     volume = {43},
     number = {6},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2004_43_6_a5/}
}
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V. G. Puzarenko. The Löwenheim–Skolem–Mal'tsev Theorem for $\mathbb{HF}$-Structures. Algebra i logika, Tome 43 (2004) no. 6, pp. 749-758. http://geodesic.mathdoc.fr/item/AL_2004_43_6_a5/