Three remarks on the modular commutator

First a problem of Ralph McKenzie is answered by proving that in a nitely directly representable variety every directly indecomposable algebra must be nite. Then we show that there is no local proof of the fundamental theorem of Abelian algebras nor of H. P. Gumm's permutability results. This part may also be of interest for those investigating non-modular Abelian algebras. Finally we provide a Gumm type-characterization of the situation when two not necessarily comparable congruences centralize each other. In doing this, we introduce a four variable version of the diierence term in every modular variety. A \two-terms condition" is also investigated. We assume that the reader is familiar with the basics of commutator theory, the two main introductory references are R. Freese and R. McKenzie 3] and H. P. Gumm 4]. Joins in lattices are denoted by +, meets by or by juxtaposition. 1. Infinite directly indecomposable algebras If a locally nite variety V has only nitely many directly indecomposable algebras among its nite members, then it has a very nice structure. Such varieties are called nitely directly representable, or FDR for short. Theorem 1.1. (R. McKenzie 8]) Let V be a FDR variety. Then V is congruence per-mutable, every nite directly indecomposable member of V is either simple or Abelian, and V satisses the commutator identity x; y] = x y 1; 1] (called C2). Every subdirectly irreducible algebra of V is nite. Hence, every nite algebra in a FDR variety can be decomposed into a direct product of neutral simple algebras and an Abelian algebra. The question how the \Abelian part" looks like can be reduced to a problem about nite rings, which has been investigated earlier by ring theorists (see 8] for details). Therefore we can say that the structure of nite algebras in FDR varieties is well understood modulo ring theory. However, no result has been obtained so far about the structure of innnite algebras in such varieties. What are the conditions on a FDR variety in order that it have no innnite directly indecomposable members? The answer is this.