On some filters and ideals of the Medvedev lattice
Let \(M\) be the Medvedev lattice of degrees of difficulty. This paper investigates some filters and ideals of \(M\). Given a property \(P\) of mass problems (i.e. of sets of functions of one variable from the set of natural numbers into itself), let us say that a degree of difficulty \({\mathfrak a}\) has property \(P\), if \({\mathfrak a}\) contains a mass problem having that property. Amongst the ideals and filters of \(M\) considered here are: the ideal generated by the dense degrees; the filter generated by the non-zero finite degrees; the filter generated by the non-zero discrete degrees; the filter generated by the non-zero countable degrees; the filter generated by the non-zero degrees with countable basis; the filter generated by the non-zero degrees of enumerability. Most of these filters and ideals were introduced by \textit{E. Z. Dyment} [Mat. Sb., Nov. Ser. 101(143), 360--379 (1976; Zbl 0353.02019)]. None of these filters and ideals is principal. It is shown in the paper that none of these filters is prime, whereas the ideal generated by the dense degrees is prime. Dividing \(M\) by any of these filters, we get a quotient lattice, whose cardinality is shown to be the same as the cardinality of \(M\). Some facts about the structure of the Turing degrees and the partial degrees are occasionally pointed out. For instance, it is shown that any non-zero partial degree bounds a quasi-minimal partial degree.
Medvedev lattice of degrees of difficulty
scientific article; zbMATH DE number 4154427