Commutator theory without join-distributivity

Author:
Paolo Lipparini

Journal:
Trans. Amer. Math. Soc. **346** (1994), 177-202

MSC:
Primary 08B10; Secondary 08A30, 08B05

DOI:
https://doi.org/10.1090/S0002-9947-1994-1257643-7

MathSciNet review:
1257643

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Abstract | References | Similar Articles | Additional Information

Abstract: We develop Commutator Theory for congruences of general algebraic systems (henceforth called algebras) assuming only the existence of a ternary term such that , whenever is a congruence and .

Our results apply in particular to congruence modular and -permutable varieties, to most locally finite varieties, and to inverse semigroups.

We obtain results concerning permutability of congruences, abelian and solvable congruences, connections between congruence identities and commutator identities. We show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis. For other lattices, some intervals are forced to be abelian, and others are forced to be nonabelian.

We give simplified proofs of some results about the commutator in modular varieties, and generalize some of them to single algebras having a modular congruence lattice.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1257643-7

Keywords:
Commutator,
congruence lattice,
difference term,
congruence identity,
abelian,
solvable,
permutable

Article copyright:
© Copyright 1994
American Mathematical Society