Notre Dame Journal of Formal Logic

Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras

Oleg V. Belegradek


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.

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Notre Dame J. Formal Logic, Volume 37, Number 4 (1996), 613-624.

First available in Project Euclid: 16 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
Secondary: 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] 03D40: Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15] 08C15: Quasivarieties 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70]


Belegradek, Oleg V. Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras. Notre Dame J. Formal Logic 37 (1996), no. 4, 613--624. doi:10.1305/ndjfl/1040046145.

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