Maximal set

Abstract: The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

Abstract: Our principal result is that there exist two incomparable recursively enumerable degrees whose greatest lower bound in the upper semilattice of degrees is 0. This was conjectured by Sacks [5]. As a secondary result, we prove that on the other hand there exists a recursively enumerable degree a < 0(1) such that for no non-zero recursively enumerable degree b is 0 the greatest lower bound of a and b. The proof of the main theorem involves a method that we have developed elsewhere [8] to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority. A different method of handling such problems has been previously introduced by Sacks [5] and used by him to prove that the recursively enumerable degrees are dense [6]. In the definition of two recursively enumerable degrees which are merely incomparable, as in the original papers of Friedberg [1] and Mucnik [3], each partial recursive functional eventually ceases to interfere with the construction, although there is no effective procedure for deciding at which state in the construction this occurs. Our second theorem provides another variation on this theme. ' All the necessary background material may be found in [2] and [5]. If e is a number and A is a set, then we define the partial function (' by setting: