Geodesic
Elementary Theories for Rogers Semilattices
Algebra i logika, Tome 44 (2005) no. 3, pp. 261-268.

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It is proved that for every level of the arithmetic hierarchy, there exist infinitely many families of sets with pairwise non-elementarily equivalent Rogers semilattices.
Keywords: arithmetic hierarchy, Rogers semilattice, elementary theory. 
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S. A. Badaev; S. S. Goncharov; A. Sorbi. Elementary Theories for Rogers Semilattices. Algebra i logika, Tome 44 (2005) no. 3, pp. 261-268. http://geodesic.mathdoc.fr/item/AL_2005_44_3_a0/

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