Commutator theory without joindistributivity
Author:
Paolo Lipparini
Journal:
Trans. Amer. Math. Soc. 346 (1994), 177202
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 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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 S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Math., 78, Springer, New York, 1981. MR 648287 (83k:08001)
 [DF]
 A. Day and R. Freese, A characterization of identities implying congruence modularity. I, Canad. J. Math. 32 (1980), 11491167. MR 596102 (82b:08009)
 [DG]
 A. Day and H. P. Gumm, Some characterizations of the commutator, Algebra Universalis 29 (1992), 6178. MR 1145556 (93g:08005)
 [FLT]
 R. Freese, W. A. Lampe, and W. Taylor, Congruence lattices of algebras of fixed similarity type. I, Pacific J. Math. 82 (1979), 5968. MR 549832 (81d:08004)
 [FMK]
 R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc. Lecture Notes No. 125, London Math. Soc., 1987. MR 909290 (89c:08006)
 [FN]
 R. Freese and J. B. Nation, Congruence lattices of semilattices, Pacific J. Math. 49 (1973), 5158. MR 0332590 (48:10916)
 [Gu]
 H. P. Gumm, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. No. 286 (1983). MR 714648 (85e:08012)
 [He]
 C. Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged) 41 (1979), 119125. MR 534504 (80h:08011)
 [HH]
 J. Hagemann and C. Herrmann, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel) 32 (1979), 234245. MR 541622 (80j:08006)
 [HMK]
 D. Hobby and R. McKenzie, The structure of finite algebras, Contemp. Math., vol. 76, Amer. Math. Soc., Providence, R.I., 1988. MR 958685 (89m:08001)
 [Jo]
 B. Jonsson, Congruence varieties, Algebra Universalis 10 (1980), 355394. MR 564122 (81e:08004)
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 K. A. Kearnes, An ordertheoretic property of the commutator, preprint. MR 1250248 (95c:08002)
 [KMK]
 K. A. Kearnes and R. McKenzie, Commutator theory for relatively modular quasivarieties, Trans. Amer. Math. Soc. 331 (1992), 465502. MR 1062872 (92h:08004)
 [Ki]
 E. W. Kiss, Three remarks on the modular commutator, Algebra Universalis 29 (1992), 455476. MR 1201171 (93m:08002)
 [KP]
 E. W. Kiss and P. Pröhle, Problems and results in tame congruence theory. A survey of the '88 Budapest Workshop, Algebra Universalis 29 (1992), 151171. MR 1157431 (93g:08004)
 [Lp]
 P. Lipparini, permutable varieties satisfy nontrivial congruence identities, Algebra Universalis (to appear). MR 1318980 (96c:08010)
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 , Varieties satisfying some form of the Herrmann Theorem (in preparation).
 [MK]
 R. McKenzie, Some interactions between group theory and the general theory of algebras, Groups, Canberra 1989 (L. G. Kovacs, ed.), Lecture Notes in Math., vol. 1456, Springer, New York, 1990. MR 1092221 (92a:08009)
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 R. McKenzie, G. McNulty, and W. Taylor, Algebras, lattices, varieties, vol. I, Wadsworth & Brooks/Cole, Monterey, Calif., 1987. MR 883644 (88e:08001)
 [Pe]
 M. Petrich, Inverse semigroups, Wiley, 1984. MR 752899 (85k:20001)
 [Qu]
 R. Quackenbush, Quasiaffine algebras, Algebra Universalis 20 (1985), 318327. MR 811692 (87d:08006)
 [Sm]
 J. D. H. Smith, Mal'cev varieties, Lecture Notes in Math., vol. 554, Springer, 1976. MR 0432511 (55:5499)
 [Ta]
 W. Taylor, Some applications of the term condition, Algebra Universalis 14 (1982), 1124. MR 634412 (83d:08004)
 [Ta1]
 W. Taylor, Characterizing Mal'cev conditions, Algebra Universalis 3 (1973), 351397. MR 0349537 (50:2030)
 [Wi]
 R. Willard, as 0, sublattice of does not force the term condition, Proc. Amer. Math. Soc. 104 (1988), 349356. MR 962797 (90a:08002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412576437
PII:
S 00029947(1994)12576437
Keywords:
Commutator,
congruence lattice,
difference term,
congruence identity,
abelian,
solvable,
permutable
Article copyright:
© Copyright 1994 American Mathematical Society
