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 (1) the Higman Theorem implies the Generalized Higman Theorem; (2) 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.
"Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras." Notre Dame J. Formal Logic 37 (4) 613 - 624, Fall 1996. https://doi.org/10.1305/ndjfl/1040046145