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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Commutator theory without join-distributivity

Author: Paolo Lipparini
Journal: Trans. Amer. Math. Soc. 346 (1994), 177-202
MSC: Primary 08B10; Secondary 08A30, 08B05
MathSciNet review: 1257643
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Abstract: We develop Commutator Theory for congruences of general algebraic systems (henceforth called algebras) assuming only the existence of a ternary term $ d$ such that $ d(a,b,b)[\alpha ,\alpha ]a[\alpha ,\alpha ]d(b,b,a)$, whenever $ \alpha $ is a congruence and $ a\alpha b$.

Our results apply in particular to congruence modular and $ n$-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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Keywords: Commutator, congruence lattice, difference term, congruence identity, abelian, solvable, permutable
Article copyright: © Copyright 1994 American Mathematical Society