Geodesic
Local Structure of Rogers Semilattices of~$\Sigma^0_n$-Computable Numberings
Algebra i logika, Tome 44 (2005) no. 2, pp. 148-172.

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We deal in specific features of the algebraic structure of Rogers semilattices of $\Sigma^0_n$ – computable numberings, for $n\geqslant2$. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non $0'$-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.
Keywords: Rogers semilattice, Lachlan semilattice, $\Sigma^0_n$-computable numbering. 
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S. Yu. Podzorov. Local Structure of Rogers Semilattices of~$\Sigma^0_n$-Computable Numberings. Algebra i logika, Tome 44 (2005) no. 2, pp. 148-172. http://geodesic.mathdoc.fr/item/AL_2005_44_2_a1/

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