Determining Whether V(A) Has a Model Companion is Undecidable

@article{Willard2004DeterminingWV,
  title={Determining Whether V(A) Has a Model Companion is Undecidable},
  author={Ross Willard},
  journal={Int. J. Algebra Comput.},
  year={2004},
  volume={14},
  pages={325-355}
}
  • R. Willard
  • Published 1 June 2004
  • Mathematics
  • Int. J. Algebra Comput.
Using techniques pioneered by R. McKenzie, we prove that there is no algorithm which, given a finite algebra in a finite language, determines whether the variety (equational class) generated by the algebra has a model companion. In particular, there exists a finite algebra such that the variety it generates has no model companion; this answers a question of Burris and Werner from 1979. 
An overview of modern universal algebra
Abstract. This article, aimed specifically at mathematical logicians, gives a gentle introduction to some of the central achievements and problems of universal algebra during the last 25 years. I

References

SHOWING 1-10 OF 16 REFERENCES
A Residually Small, Finitely Generated, Semi-Simple Variety which is not Residually Finite
TLDR
An 8 element algebra A′ is described which generates a semi-simple variety which contains a countably infinite simple algebra into which all other infinite simple algebras in the variety can be embedded.
The Residual Bounds of Finite Algebras
TLDR
An eight-element simple algebra with eight operations that is inherently non-finitely-based and generates a precisely residually countable variety.
The Residual Bound of a Finite Algebra is not Computable
  • R. McKenzie
  • Mathematics, Computer Science
    Int. J. Algebra Comput.
  • 1996
TLDR
A construction is exhibited which produces for every Turing machine, an algebra such that the Turing machine halts iff the variety generated by is residually finite.
Sheaf constructions and their elementary properties
We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability
On Mckenzie's method
This is an expository account of R. McKenzie's recent refutation of the RS conjecture.
A course in universal algebra
Model-completions and modules
Model theory
  • W. Hodges
  • Computer Science
    Encyclopedia of mathematics and its applications
  • 1993
By Lemmas 2.3 and 9
  • By Lemmas 2.3 and 9
Theorems 7.1 and 9.7, the recursive unsolvability of the halting problem, and the effectiveness of the construction of M(T ) from T
  • Theorems 7.1 and 9.7, the recursive unsolvability of the halting problem, and the effectiveness of the construction of M(T ) from T
...
...