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.
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Notes
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This understanding of implication departs from the traditional concept according to which implication is viewed as a sentential connective conjoining two sentential formulas and thereby yielding a new formula. AAL provides a more general meaning of the term “implication”. In particular, in equational logic, an implication (without parameters) is any set \(\Rightarrow \) of quaternary equations having the detachment property with respect to a given consequence operation \(\mathbf{Q}^{\models }\).
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Czelakowski, J. (2015). Introduction. In: The Equationally-Defined Commutator. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-21200-5_1
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