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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. 1991 Mathematics Subject Classification. 03C05.doi:10.1142/s0218196704001748 fatcat:qqoa3wi2jffwlhz7au5nlqtlxe