## Notre Dame Journal of Formal Logic

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

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

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 16 December 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1040046145

**Digital Object Identifier**

doi:10.1305/ndjfl/1040046145

**Mathematical Reviews number (MathSciNet)**

MR1446232

**Zentralblatt MATH identifier**

0882.03036

**Subjects**

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]

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1040046145