DETERMINING WHETHER ${\mathsf V}({\bf A})$ HAS A MODEL COMPANION IS UNDECIDABLE
Ross Willard · International Journal of Algebra and Computation · 2004
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.