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Finite basis problem for varieties of algebraic systems

Vesselin Drensky

Abstract

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with the property that they and their subvarieties are finitely based. A special attention is paid on the varieties of semigroups, groups, associative, Lie and other nonassociative algebras.

Finite basis problem for varieties of algebraic systems

Abstract

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with the property that they and their subvarieties are finitely based. A special attention is paid on the varieties of semigroups, groups, associative, Lie and other nonassociative algebras.
Paper Structure (30 sections, 55 theorems, 94 equations)

This paper contains 30 sections, 55 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Structure theory of algebraic systems
  3. Combinatorial approach to the theory of algebraic systems
  4. Varieties of algebraic systems
  5. Trivial answer: Infinitely based varieties of nonassociative objects
  6. Meeting points with mathematical logic
  7. Positive results
  8. Structure theory of groups and algebras
  9. The Higman-Cohen method
  10. Methods of commutative algebra
  11. Weak polynomial identities
  12. Structure theory of T-ideals -- the approach of Kemer
  13. Finite algebraic system with a finite system of operations
  14. Infinitely based varieties of semigroups
  15. The first examples
  16. ...and 15 more sections

Key Result

Theorem 1.4

(HSP-Theorem of Birkhoff Birk1) (i) The class $\mathfrak W$ of algebraic systems from $\mathfrak V$ is a variety if and only if it is closed under homomorphic objects ($\mathcal{H}$), subobjects ($\mathcal{S}$), and cartesian products ($\mathcal{P}$). (ii) The variety ${\mathfrak W}=\text{var}(A)$ i

Theorems & Definitions (60)

  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 2.1
  • Theorem 3.3
  • Theorem 4.1
  • Definition 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 50 more