The Relationship Between Two Commutators

We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi-affine in such varieties. We refine this by showing that if A is an abelian algebra and (A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.

[1]  Gábor Czédli.  A characterization for congruence semi-distributivity , 1983 .

[2]  Some characterizations of the commutator , 1992 .

[3]  W. Taylor.  Varieties Obeying Homotopy Laws , 1977, Canadian Journal of Mathematics.

[4]  Walter Taylor,et al.  The Lattice of Interpretability Types of Varieties , 1984 .

[5]  Every idempotent plain algebra generates a minimal variety , 1988 .

[6]  Local Malcev Conditions , 1972, Canadian Mathematical Bulletin.

[7]  Jonathan D. H. Smith.  Mal'cev Varieties , 1977 .

[8]  Keith A. Kearnes.  Varieties with a Difference Term , 1995 .

[9]  M. Pedicchio.  A Categorical Approach to Commutator Theory , 1995 .

[10]  Paolo Lipparini.  A Characterization of Varieties with a Difference Term , 1996, Canadian Mathematical Bulletin.

[11]  James B. Nation,et al.  Problems and Results in Tame Congruence Theory a Survey of the '88 Budapest Workshop , 1992 .

[12]  R. McKenzie,et al.  Commutator theory for relatively modular quasivarieties , 1992 .

[13]  C. A. Wolf.  Many-sorted algebras in congruence modular varieties , 1996 .

[14]  Christian Herrmann,et al.  A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity , 1979 .

[15]  D. Hobby,et al.  The structure of finite algebras , 1988 .

[16]  R. McKenzie,et al.  Commutator theory for congruence modular varieties , 1989 .

[17]  James B. Nation,et al.  CONGRUENCE LATTICES OF SEMILATTICES , 1973 .

[18]  Paolo Lipparini.  n-Permutable varieties satisfy non trivial congruence identities , 1995 .

[19]  R. Quackenbush.  Quasi-affine algebras , 1985 .

[20]  H. Peter Gumm,et al.  Geometrical methods in congruence modular algebras , 1983 .