Abstract
This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow.
The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK).
A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.
Similar content being viewed by others
References
S. Burris,A note on directly indecomposable algebras, Preprint (1979).
S. Burris,Arithmetical varieties and Boolean product representations, Preprint (1978).
S. Burris andR. McKenzie,Decidability and Boolean representations, Preprint (1980).
D. Clark andP. Krauss,Para primal algebras, Alg. Univ.6 (1976), 165–192.
D. Clark andP. Krauss,Varieties generated by para primal algebras, Alg. Univ.7 (1977), 93–114.
D. Clark andP. Krauss,Plain para primal algebras, Alg. Univ. (to appear).
V. Dlab andC. M. Ringel,On algebras of finite representation type, J. Algebra33 (1975), 306–394.
V. Dlab andC. M. Ringel,Indecomposable representations of graphs and algebras, Memoirs A.M.S. 6, number173 (1976), 57 pages.
R. Freese andR. McKenzie,Residually small varieties with modular congruence lattices, Trans. Amer. Math. Soc. 264 (1981), 419–430.
R. Freese andR. McKenzie,The commutator, an overview, Preprint (1981).
G. Gratzer,A characterization of neutral elements in lattices, Publications of the Math. Institute of the Hungarian Academy of Sciences,7 (1962), 191–192.
H. P. Gumm,Algebras in permutable varieties: geometrical properties of affine algebras, Alg. Univ.9 (1979), 8–34.
J. Hagemann andC. Herrmann,A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel)32 (1979), 234–245.
C. Herrmann,Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged),41 (1979), 119–125.
R. McKenzie,On minimal, locally finite varieties with permuting congruence relations (unpublished).
R. McKenzie,Para primal varieties: A study of finite aziomatizability and definable principal congruences in locally finite varieties, Alg. Univ.8 (1978), 336–348.
R. McKenzie,Residually small varieties of K-algebras, Alg. Univ.14 (1982), 181–196.
R. Quackenbush,Algebras with small fine spectrum (unpublished).
R. Quackenbush,Algebras with minimal spectrum, Alg. Univ.10 (1980), 117–129.
R. Quackenbush,A new proof of Rosenberg's primal algebra characterization theorem, Preprint (1980).
J. D. H. Smith,Malcev Varieties, Springer Lecture Notes 554 (Berlin, 1976).
W. Taylor,Uniformity of congruences, Alg. Univ.4 (1974), 342–360.
W. Taylor,The fine spectrum of a variety, Alg. Univ.5 (1975), 263–303.
W. R. Scott,Group Theory, Prentice-Hall, New Jersey (1974).
Author information
Authors and Affiliations
Additional information
Research supported by U.S. National Science Foundation grant number MCS 77-22913.
Rights and permissions
About this article
Cite this article
McKenzie, R. Narrowness implies uniformity. Algebra Universalis 15, 67–85 (1982). https://doi.org/10.1007/BF02483709
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02483709