## A finite basis theorem for difference-term varieties with a finite residual bound

HTML articles powered by AMS MathViewer

- by Keith Kearnes, Ágnes Szendrei and Ross Willard PDF
- Trans. Amer. Math. Soc.
**368**(2016), 2115-2143 Request permission

## Abstract:

We prove that if $\mathcal V$ is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, $\mathcal V$ has a difference term, and $\mathcal V$ has a finite residual bound, then $\mathcal V$ is finitely axiomatizable. This provides a common generalization of R. McKenzie’s finite basis theorem for congruence modular varieties with a finite residual bound, and R. Willard’s finite basis theorem for congruence meet-semidistributive varieties with a finite residual bound.## References

- Kirby A. Baker,
*Equational bases for finite algebras*, Notices Amer. Math. Soc.**19**(1972), A–44, 691-08-02. - Kirby A. Baker,
*Finite equational bases for finite algebras in a congruence-distributive equational class*, Advances in Math.**24**(1977), no. 3, 207–243. MR**447074**, DOI 10.1016/0001-8708(77)90056-1 - Kirby A. Baker, George F. McNulty, and Ju Wang,
*An extension of Willard’s finite basis theorem: congruence meet-semidistributive varieties of finite critical depth*, Algebra Universalis**52**(2004), no. 2-3, 289–302. MR**2161654**, DOI 10.1007/s00012-004-1890-0 - Garrett Birkhoff,
*On the structure of abstract algebras*, Proc. Camb. Philos. Soc.**31**(1935), 433–454. - Roger M. Bryant,
*The laws of finite pointed groups*, Bull. London Math. Soc.**14**(1982), no. 2, 119–123. MR**647192**, DOI 10.1112/blms/14.2.119 - Gábor Czédli,
*A characterization for congruence semidistributivity*, Universal algebra and lattice theory (Puebla, 1982) Lecture Notes in Math., vol. 1004, Springer, Berlin, 1983, pp. 104–110. MR**716177**, DOI 10.1007/BFb0063432 - Ralph Freese and Ralph McKenzie,
*Residually small varieties with modular congruence lattices*, Trans. Amer. Math. Soc.**264**(1981), no. 2, 419–430. MR**603772**, DOI 10.1090/S0002-9947-1981-0603772-9 - Ralph Freese and Ralph McKenzie,
*Commutator theory for congruence modular varieties*, London Mathematical Society Lecture Note Series, vol. 125, Cambridge University Press, Cambridge, 1987. MR**909290** - H.-Peter Gumm (ed.),
*Workshop report*, Mathematisches Forschungsinstitut Oberwolfach, 1976, workshop no. 35 held Aug. 15–21, unpublished. - H.-Peter Gumm,
*Congruence modularity is permutability composed with distributivity*, Arch. Math. (Basel)**36**(1981), no. 6, 569–576. MR**629294**, DOI 10.1007/BF01223741 - H. Peter Gumm,
*Geometrical methods in congruence modular algebras*, Mem. Amer. Math. Soc.**45**(1983), no. 286, viii+79. MR**714648**, DOI 10.1090/memo/0286 - David Hobby and Ralph McKenzie,
*The structure of finite algebras*, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR**958685**, DOI 10.1090/conm/076 - I. M. Isaev,
*Inherently non-finitely based varieties of algebras*, Sibirsk. Mat. Zh.**30**(1989), no. 6, 75–77 (Russian); English transl., Siberian Math. J.**30**(1989), no. 6, 892–894 (1990). MR**1043435**, DOI 10.1007/BF00970911 - Bjarni Jónsson,
*Algebras whose congruence lattices are distributive*, Math. Scand.**21**(1967), 110–121 (1968). MR**237402**, DOI 10.7146/math.scand.a-10850 - Bjarni Jónsson and Ivan Rival,
*Lattice varieties covering the smallest nonmodular variety*, Pacific J. Math.**82**(1979), no. 2, 463–478. MR**551703** - Keith A. Kearnes,
*An order-theoretic property of the commutator*, Internat. J. Algebra Comput.**3**(1993), no. 4, 491–533. MR**1250248**, DOI 10.1142/S0218196793000299 - Keith A. Kearnes,
*Varieties with a difference term*, J. Algebra**177**(1995), no. 3, 926–960. MR**1358491**, DOI 10.1006/jabr.1995.1334 - Keith A. Kearnes,
*Cardinality bounds for subdirectly irreducible algebras*, J. Pure Appl. Algebra**112**(1996), no. 3, 293–312. MR**1410180**, DOI 10.1016/0022-4049(95)00137-9 - K. A. Kearnes,
*A Hamiltonian property for nilpotent algebras*, Algebra Universalis**37**(1997), no. 4, 403–421. MR**1465297**, DOI 10.1007/s000120050025 - Keith A. Kearnes and Ágnes Szendrei,
*The relationship between two commutators*, Internat. J. Algebra Comput.**8**(1998), no. 4, 497–531. MR**1663558**, DOI 10.1142/S0218196798000247 - Emil W. Kiss,
*Three remarks on the modular commutator*, Algebra Universalis**29**(1992), no. 4, 455–476. MR**1201171**, DOI 10.1007/BF01190773 - Emil W. Kiss,
*An easy way to minimal algebras*, Internat. J. Algebra Comput.**7**(1997), no. 1, 55–75. MR**1428329**, DOI 10.1142/S021819679700006X - Robert L. Kruse,
*Identities satisfied by a finite ring*, J. Algebra**26**(1973), 298–318. MR**325678**, DOI 10.1016/0021-8693(73)90025-2 - Paolo Lipparini,
*Commutator theory without join-distributivity*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 177–202. MR**1257643**, DOI 10.1090/S0002-9947-1994-1257643-7 - P. Lipparini,
*$n$-permutable varieties satisfy nontrivial congruence identities*, Algebra Universalis**33**(1995), no. 2, 159–168. MR**1318980**, DOI 10.1007/BF01190927 - Paolo Lipparini,
*A characterization of varieties with a difference term*, Canad. Math. Bull.**39**(1996), no. 3, 308–315. MR**1411074**, DOI 10.4153/CMB-1996-038-0 - Paolo Lipparini,
*A characterization of varieties with a difference term. II. Neutral $=$ meet semi-distributive*, Canad. Math. Bull.**41**(1998), no. 3, 318–327. MR**1637665**, DOI 10.4153/CMB-1998-044-9 - P. Lipparini,
*A Kiss $4$-difference term from a ternary term*, Algebra Universalis**42**(1999), no. 1-2, 153–154. MR**1736348**, DOI 10.1007/s000120050130 - I. V. L′vov,
*Varieties of associative rings. I, II*, Algebra i Logika**12**(1973), 269–297, 363; ibid. 12 (1973), 667–688, 735 (Russian). MR**0389973** - I. V. L′vov,
*Finite-dimensional algebras with infinite identity bases*, Sibirsk Mat. Ž.**19**(1978), no. 1, 91–99, 237 (Russian). MR**0506540** - R. C. Lyndon,
*Identities in two-valued calculi*, Trans. Amer. Math. Soc.**71**(1951), 457–465. MR**44470**, DOI 10.1090/S0002-9947-1951-0044470-3 - R. C. Lyndon,
*Identities in finite algebras*, Proc. Amer. Math. Soc.**5**(1954), 8–9. MR**60482**, DOI 10.1090/S0002-9939-1954-0060482-6 - Miklós Maróti and Ralph McKenzie,
*Finite basis problems and results for quasivarieties*, Studia Logica**78**(2004), no. 1-2, 293–320. MR**2108031**, DOI 10.1007/s11225-005-3320-5 - Ralph McKenzie,
*Equational bases for lattice theories*, Math. Scand.**27**(1970), 24–38. MR**274353**, DOI 10.7146/math.scand.a-10984 - Ralph McKenzie,
*Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties*, Algebra Universalis**8**(1978), no. 3, 336–348. MR**469853**, DOI 10.1007/BF02485404 - Ralph McKenzie,
*Finite equational bases for congruence modular varieties*, Algebra Universalis**24**(1987), no. 3, 224–250. MR**931614**, DOI 10.1007/BF01195263 - Ralph McKenzie,
*Tarski’s finite basis problem is undecidable*, Internat. J. Algebra Comput.**6**(1996), no. 1, 49–104. MR**1371734**, DOI 10.1142/S0218196796000040 - George F. McNulty and Caroline R. Shallon,
*Inherently nonfinitely based finite algebras*, Universal algebra and lattice theory (Puebla, 1982) Lecture Notes in Math., vol. 1004, Springer, Berlin, 1983, pp. 206–231. MR**716184**, DOI 10.1007/BFb0063439 - Sheila Oates and M. B. Powell,
*Identical relations in finite groups*, J. Algebra**1**(1964), 11–39. MR**161904**, DOI 10.1016/0021-8693(64)90004-3 - Robert Edward Park,
*EQUATIONAL CLASSES OF NON-ASSOCIATIVE ORDERED ALGEBRAS*, ProQuest LLC, Ann Arbor, MI, 1976. Thesis (Ph.D.)–University of California, Los Angeles. MR**2626431** - Peter Perkins,
*Bases for equational theories of semigroups*, J. Algebra**11**(1969), 298–314. MR**233911**, DOI 10.1016/0021-8693(69)90058-1 - S. V. Polin,
*Identities of finite algebras*, Sibirsk. Mat. Ž.**17**(1976), no. 6, 1356–1366, 1439 (Russian). MR**0439715** - Robert W. Quackenbush,
*Equational classes generated by finite algebras*, Algebra Universalis**1**(1971/72), 265–266. MR**294222**, DOI 10.1007/BF02944989 - Walter Taylor,
*Equational logic*, Contributions to universal algebra (Colloq., József Attila Univ., Szeged, 1975) Colloq. Math. Soc. János Bolyai, Vol. 17, North-Holland, Amsterdam, 1977, pp. 465–501. MR**0472645** - Ross Willard,
*A finite basis theorem for residually finite, congruence meet-semidistributive varieties*, J. Symbolic Logic**65**(2000), no. 1, 187–200. MR**1782114**, DOI 10.2307/2586531 - Ross Willard,
*The finite basis problem*, Contributions to general algebra. 15, Heyn, Klagenfurt, 2004, pp. 199–206. MR**2082383**

## Additional Information

**Keith Kearnes**- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado at Boulder, Boulder, Colorado 80309-0385
- MR Author ID: 99640
**Ágnes Szendrei**- Affiliation: Department of Mathematics, Campus Box 395, University of Colorado at Boulder, Boulder, Colorado 80309-0385 – and – Bolyai Institute, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
**Ross Willard**- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Received by editor(s): October 12, 2013
- Received by editor(s) in revised form: June 3, 2014
- Published electronically: July 10, 2015
- Additional Notes: This material is based upon work supported by the Hungarian National Foundation for Scientific Research (OTKA) grants No. K83219 and K104251 and by the Natural Sciences and Engineering Research Council of Canada.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 2115-2143 - MSC (2010): Primary 03C05; Secondary 08B05
- DOI: https://doi.org/10.1090/tran/6509
- MathSciNet review: 3449235