Logically-geometrical similarity for algebras and models with the same identities. (English) Zbl 1506.08010
Summary: The paper is related to the field which we call Universal Algebraic Geometry (UAG). All algebras under consideration belong to a variety of algebras \(\Theta\). For an arbitrary \(\Theta\) we construct a system of notions which lead to a bunch of new problems. As a rule, their solutions depend on the choice of specific \(\Theta\). It can be the variety of groups \(Grp\), the variety of associative or Lie algebras, etc. In particular, it can be the classical variety \(Com-P\) of commutative and associative algebras with a unit over a field.
For example, the paper concerns with the following general problem. For every algebra \(H\in\Theta\) one can define the category of algebraic sets over \(H\). Given \(H_1\) and \(H_2\) in \(\Theta\), the question is what are the relations between these algebras that provide an isomorphism of the corresponding categories of algebraic sets. Similar problem stands with respect to the situation when algebras are replaced by models and categories of algebraic sets are replaced by categories of definable sets. The results on the stated problem are applicable to knowledge theory and, in particular, to knowledge bases.
For example, the paper concerns with the following general problem. For every algebra \(H\in\Theta\) one can define the category of algebraic sets over \(H\). Given \(H_1\) and \(H_2\) in \(\Theta\), the question is what are the relations between these algebras that provide an isomorphism of the corresponding categories of algebraic sets. Similar problem stands with respect to the situation when algebras are replaced by models and categories of algebraic sets are replaced by categories of definable sets. The results on the stated problem are applicable to knowledge theory and, in particular, to knowledge bases.
MSC:
| 08C05 | Categories of algebras |
| 03G27 | Abstract algebraic logic |
| 14A99 | Foundations of algebraic geometry |
References:
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