Residually small varieties with modular congruence lattices

Authors:
Ralph Freese and Ralph McKenzie

Journal:
Trans. Amer. Math. Soc. **264** (1981), 419-430

MSC:
Primary 08B10; Secondary 06B10

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603772-9

MathSciNet review:
603772

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We focus on varieties of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety is residually small, then the following law holds identically for congruences over algebras in . (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety is residually small if and only if it satisfies this commutator identity, and in that case is actually residually for some finite integer . It is further proved that in a modular variety generated by a finite algebra the chief factors of any finite algebra are bounded in cardinality by the size of , and every simple algebra in the variety has a cardinality at most that of .

**[1]**J. T. Baldwin,*The number of subdirectly irreducible algebras in a variety*. II, Algebra Universalis (to appear). MR**593008 (82c:08008)****[2]**J. T. Baldwin and J. Berman,*The number of subdirectly irreducible algebras in a variety*, Algebra Universalis**5**(1975), 379-389. MR**0392765 (52:13578)****[3]**P. Crawley and R. Dilworth,*Algebraic theory of lattices*, Prentice-Hall, Englewood Cliffs, N. J., 1973.**[4]**V. Dlab and C. M. Ringel,*On algebras of finite representation type*, J. Algebra**33**(1975), 306-394. MR**0357506 (50:9974)****[5]**R. Freese and B. Jónsson,*Congruence modularity implies the Arguesian identity*, Algebra Universalis**6**(1976), 225-228. MR**0472644 (57:12340)****[6]**H. P. Gumm,*Über die Lösungsmengen von Gleichungssystemen über allgemeinen Algebren*, Math. Z.**162**(1978), 51-62. MR**505916 (80c:08001)****[7]**-,*Algebras in permutable varieties: geometrical properties of affine algebras*, Algebra Universalis**9**(1979), 8-34. MR**508666 (80d:08010)****[8]**H. P. Gumm and C. Herrmann,*Algebras in modular varieties: Baer refinements, cancellation and isotopy*, Houston J. Math.**5**(1979), 503-523. MR**567909 (81h:08008)****[9]**J. Hagemann and C. Herrmann,*A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity*, Arch. Math.**32**(1979), 234-245. MR**541622 (80j:08006)****[10]**C. Herrmann,*Affine algebras in congruence modular varieties*, Acta Sci. Math.**41**(1979), 119-125. MR**534504 (80h:08011)****[11]**L. G. Kovács and M. F. Newman,*On critical groups*, J. Austral. Math. Soc.**6**(1966), 237-250. MR**0200337 (34:233)****[12]**W. A. Lampe and W. Taylor,*Simple algebras in varieties*(preprint). MR**634414 (83e:08012)****[13]**R. McKenzie,*Residually small varieties of semigroups*, Algebra Universalis (to appear). MR**631555 (82k:20097)****[14]**R. McKenzie and S. Shelah,*The cardinals of simple models for universal theories*, Proc. Sympos. Pure Math., vol. 25, Amer. Math. Soc., Providence, R. I., 1974, pp. 53-74. MR**0360261 (50:12711)****[15]**S. Oates and M. B. Powell,*Identical relations infinite groups*, J. Algebra**1**(1964), 11-39. MR**0161904 (28:5108)****[16]**A. Yu. Ol'shanskii,*Conditional identities in finite groups*, Sibirsk. Mat. Z.**15**(1974), 1409-1413. (English translation) MR**0367068 (51:3310)****[17]**R. W. Quachenbush,*Equational classes generated by finite algebras*, Algebra Universalis**1**(1971), 265-266. MR**0294222 (45:3295)****[18]**A. Rosenberg and D. Zelinsky,*Finiteness of the injective hull*, Math. Z.**70**(1959), 372-380. MR**0105434 (21:4176)****[19]**C. Shallon,*Non-finitely based algebras derived from lattices*, Ph. D. Thesis, UCLA, 1978.**[20]**J. D. H. Smith,*Mal'cev varieties*, Lecture Notes in Math., vol. 554, Springer-Verlag, Berlin, 1976. MR**0432511 (55:5499)****[21]**W. Taylor,*Residually small varieties*, Algebra Universalis**2**(1972), 33-53. MR**0314726 (47:3278)****[22]**-,*Subdirectly irreducible algebras in regular, permutable varieties*, Proc. Amer. Math. Soc.**75**(1979), 196-200. MR**532134 (80h:08010)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
08B10,
06B10

Retrieve articles in all journals with MSC: 08B10, 06B10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603772-9

Keywords:
Residually small variety,
subdirect product,
modular congruence lattice,
commutator of congruences,
chief factors of universal algebras,
groups with abelian Sylow subgroups

Article copyright:
© Copyright 1981
American Mathematical Society