On the Loewy Rank of Infinite Algebras

The Loewy rank of a complete lattice L is deened as follows. Take the meet a 1 of all coatoms of L. Then let a 2 be the meet of all lower covers of a 1. Iterate this process to deene a for every ordinal by letting a +1 be the meet of all lower covers of a , and a the meet of all a (for <) if is a limit ordinal. The Loewy rank of L is the smallest ordinal for which a is the zero of L, and the symbol 1 if such does not exist. The Loewy rank L(A) of an algebra A is deened to be the Loewy rank of Con(A), and for a class K of algebras L(K) = supfL(A) j A 2 Kg. The above concept has been introduced in 2.2 of Ralph McKenzie's paper 3]. There he proves (in Corollary 2.11) that if A is a nite algebra generating a congruence modular variety V, then the Loewy rank of any nite member of V is bounded by L(S(A)). This statement follows from two easy observations. The rst one is that the Loewy rank of a subdirect product (in a modular variety) cannot exceed the supremum of the Loewy ranks of its factors. The second observation is that L(H(A)) L(A) provided that Con(A) is nite and modular. This second statement does not hold for innnite lattices, however (take the ring of integers). Therefore the above argument does not give a bound for the innnite members of V. In 2.13 and 3.11 of 3] Ralph McKenzie asks whether L(V) is nite. We answer this question by proving Theorem. Let A be a nite algebra in a modular variety. Then L(V(A)) jAj. We shall prove a better result actually, but we were not able to show that L(V) = L(V n) for every nitely generated modular variety V. In fact, the proof below indicates that this might not be the case in general (see the discussion at the end). The idea of the proof is to extend a group theoretic lemma, found in L. G. Kovv acs, M. F. Newman 2], to the congruence modular case, and to innnite subdirectly irreducibles. In group theory, subdirectly irreducible algebras are sometimes called monolithic. The monolith of a subdirectly irreducible algebra S, that is, its smallest nonzero congruence, is denoted by (S) in this paper. Let us call …