Geodesic
On elementary and geometric equivalence of equational co-domains
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 4, pp. 229-238.

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We prove that the class of equational co-domains is not closed under elementary and geometric equivalence. We find corresponding counter-examples in predicate and functional languages. 
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A. N. Shevlyakov. On elementary and geometric equivalence of equational co-domains. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 4, pp. 229-238. http://geodesic.mathdoc.fr/item/FPM_2019_22_4_a14/

[1] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., “Algebraicheskaya geometriya nad algebraicheskimi sistemami. VIII. Geometricheskie ekvivalentnosti i osobye klassy algebraicheskikh sistem”, Fundament. i prikl. matem., 22:4 (2019), 75–100

[2] Daniyarova E. Yu., Myasnikov A. G., Remeslennikov V. N., Algebraicheskaya geometriya nad algebraicheskimi sistemami, Izd-vo SO RAN, Novosibirsk, 2016

[3] Shevlyakov A. N., Lectures notes in universal algebraic geometry, arXiv: 1601.02743