Algebraic geometry over algebraic structures. IV. Equational domains and codomains

Abstract

We introduce and study equational domains and equational codomains. Informally, an equational domain is an algebra every finite union of algebraic sets over which is an algebraic set; an equational codomain is an algebra every proper finite union of algebraic sets over which is not an algebraic set.

Equations in polyadic groups

“…Equations over Lie algebras, associative algebras and semigroups are also studied (for example, see [7], [10], and [33]). In a series of fundamental papers, Daniyarova, Myasnikov and Remeslennikov introduced the basics of universal algebraic geometry ( [6], [7], [8] and [9]). …”

“…Our notations dealing with universal algebraic geometry, are the same as [6], [7], [8], [9]. The reader should consult these references for a complete account of the universal algebraic geometry.…”

Algebraic geometry over algebraic structures. V. The case of arbitrary signature

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A general theory of algebraic geometry over an arbitrary algebraic structure A in a language L with no predicates is consistently presented in a series of papers on universal algebraic geometry [5][6][7][8]. The restriction that we impose on the language is not crucial.

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“…Therefore, it became clear that it would be more convenient to have an apparatus of universal algebraic geometry bearing on algebraic structures in an arbitrary signature too. However, the first four works [5][6][7][8] in our series of papers on algebraic geometry over algebraic structures had already been published or accepted for publication, with the material expounded in them containing the restriction on the signature. By this token, we come up with this supplement to the papers mentioned, where we describe how to pass from a signature with no predicates to the general case.…”

“…In this group, definitions are couched for the concepts of a coordinate algebra, of approximability and discriminability. Also two theorems in [8] and their corollaries fall into the second group. At the same time, making use of the opportunity, we would like to correct a little oversight that crept into one of the two theorems in [8].…”