Geodesic
Ideals without minimal elements in Rogers semilattices
Algebra i logika, Tome 54 (2015) no. 3, pp. 305-314

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We prove a criterion for the existence of a minimal numbering, which is reducible to a given numbering of an arbitrary set. The criterion is used to show that, for any infinite $A$-computable family $F$ of total functions, where $\varnothing'\le_TA$, the Rogers semilattice $\mathcal R_A(F)$ of $A$-computable numberings for $F$ contains an ideal without minimal elements.
Keywords: minimal numbering, $A$-computable numbering, Rogers semilattice, ideal. 
@article{AL_2015_54_3_a0,
     author = {A. A. Issakhov},
     title = {Ideals without minimal elements in {Rogers} semilattices},
     journal = {Algebra i logika},
     pages = {305--314},
     publisher = {mathdoc},
     volume = {54},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2015_54_3_a0/}
}
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A. A. Issakhov. Ideals without minimal elements in Rogers semilattices. Algebra i logika, Tome 54 (2015) no. 3, pp. 305-314. http://geodesic.mathdoc.fr/item/AL_2015_54_3_a0/