Categorical Horn theories and modules

Categorical horn classes. 2

“…A class K in a language L is categorical if K is categorical in some cardinality greater than |L|. According to [1], a class is categorical if it is categorical in some cardinality not less than |L|. Note that according to Morley-Shelah's theorem, a categorical axiomatizable class in a language L is categorical in all cardinalities greater than |L| (see [4]).…”

“…This paper is a continuation of [1], which was published 30 years ago. The reason for such a long time interval is that I pinned hopes on obtaining a fairly complete description of uncountably categorical Horn classes.…”

“…Likewise, a theory for a structure A is primitive normal if a theory for its Cartesian degree A ω is stable [1].…”

“…For the sake of convenience, we lay out certain of the definitions and formulations of the lemmas used in [1]. In so doing, an asterisk is ascribed to the original numbering of the results, the term a "structure" is used instead of the term an "algebraic system," and the term a "signature" is replaced by the term a "language."…”

“…The reason for such a long time interval is that I pinned hopes on obtaining a fairly complete description of uncountably categorical Horn classes. For instance, as claimed in the last section of [1], my intention was to prove that the uncountably categorical Horn classes are reducts of categorical quasivarieties. Since these quasivarieties had been described completely, this would furnish a nice description of categorical Horn classes.…”