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I will give a brief overview of Palyutin's work on Horn theories, together with a more detailed discussion of his recent paper [4]

A Horn class of structures (for the same language) is a (finitary first-order) axiomatizable class closed under filtered products. A class of structures is categorical if it has a unique member up to isomorphism in all cardinalities greater than the cardinality of the language. A positive primitive formula is some existential quantifications of a conjunction of atomic formulas.

Palyutin describes the categorical Horn classes of modules. He investigates conditions on a categorical Horn class that would imply it is equivalent (via positive primitive definitions) to a theory of modules.

All the articles cited are by E. Palyutin and the first three all appeared in Algebra and Logic

1. Description of categorical positive Horn classes, 19, No. 6, 443–455 (1981).
2. Categorical Horn classes. 1. 19, No. 5, 377–400 (1981).
3. Categorical Horn classes. 2. 49, No. 6, 526–538 (2010).
4. Categorical Horn theories and modules, Siberian Mathematical Journal, 52, No. 6, 1056–1064 (2011).