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Introduction

In: The Equationally-Defined Commutator

Author

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  • Janusz Czelakowski

    (Opole University, Institute of Mathematics and Informatics)

Abstract

Commutator theory is a part of universal algebra. It is rooted in the theories of groups and rings. From the general algebraic perspective the commutator was first investigated in the seventies by J. Smith for Mal’cev varieties. (Mal’cev varieties are characterized by the condition that all congruences on their algebras permute.) Further was done by the German algebraists H.P. Gumm, J. Hagemann and C. Herrmann in the eighties. They discovered that congruence-modular varieties (CM varieties, for short) form a natural environment for the commutator. Hagemann and Herrmann’s approach is lattice-theoretical. Gumm’s approach is based on an analogy between commutator theory and affine geometry which allowed him to discover many of the basic facts about the commutator from the geometric perspective. Freese and McKenzie (1987) summarize earlier results and establish a complementary paradigm for commutator theory in universal algebra. Kearnes and McKenzie (1992) subsequently extended the theory from congruence-modular varieties onto relatively congruence-modular quasivarieties.

Suggested Citation

  • Janusz Czelakowski, 2015. "Introduction," Springer Books, in: The Equationally-Defined Commutator, edition 1, chapter 0, pages 1-6, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-21200-5_1
    DOI: 10.1007/978-3-319-21200-5_1
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