The main result is that for every recursively enumerable existential consistent theory Γ , there exists a finitely presented SQ-universal group H such that Γ is satisfied in every nontrivial quotient of H. Furthermore if Γ is satisfied in some group with a soluble word problem, then H can be taken with a soluble word problem. We characterize the finitely generated groups with soluble word problem as the finitely generated groups G for which there exists a finitely presented group H (...) all of the nontrivial quotients of which embed G. We prove also that for every countable group G, there exists a 2-finitely generated SQ-universal group H such that every nontrivial quotient of H embeds G. (shrink)

For a quasi variety of algebras K, the Higman Theorem is said to be true if every recursively presented K-algebra is embeddable into a finitely presented K-algebra; the Generalized Higman Theorem is said to be true if any K-algebra which is recursively presented over its finitely generated subalgebra is embeddable into a K-algebra which is finitely presented over this subalgebra. We suggest certain general conditions on K under which the Higman Theorem implies the Generalized Higman Theorem; a finitely generated K-algebra (...) A is embeddable into every existentially closed K-algebra containing a finitely generated K-algebra B if and only if the word problem for A is Q-reducible to the word problem for B. The quasi varieties of groups, torsion-free groups, and semigroups satisfy these conditions. (shrink)

We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.

A splitting A1A2 = A of an r.e. set A is called a Friedberg splitting if for any r.e. set W with W — A not r.e., W — Ai≠0 for I = 1,2. In an earlier paper, the authors investigated Friedberg splittings of maximal sets and showed that they formed an orbit with very interesting degree-theoretical properties. In the present paper we continue our investigations, this time analyzing Friedberg splittings and in particular their orbits and degrees for various classes (...) of r.e. sets. (shrink)