Modular Varieties with the Fraser-horn Property

The notion of central idempotent elements in a ring can be easily generalized to the setting of any variety with the property that proper subalgebras are always nontrivial. We will prove that if such a variety is also congruence modular, then it has factorable congruences, i.e., it has the FraserHorn property. (This property is well known to have major implications for the structure theory of the algebras in the variety.) By a variety with ~0 and ~1 we understand a variety V for which there exist unary terms 01(w), ..., 0n(w), 11(w), ..., 1n(w) such that V |= ~0(w) = ~1(w) → x = y, where ~0 = (01, ..., 0n) and ~1 = (11, ..., 1n). Indeed this condition is equivalent to the more familiar property that no nontrivial algebra of V has a trivial subalgebra (combine [17, Lemma 3] with [12]). If λ ∈ A ∈ V , then we say that ~e ∈ A is a λ-central element of A if there exists an isomorphism A→ A1 ×A2 such that λ→ (λ1, λ2), ~e→ ((01(λ1), 11(λ2)), ..., (0n(λ1), 1n(λ2))) . It is well known that the central elements of a ring with identity are just the central idempotent ones. To give another example, an element in a bounded lattice is central iff it is neutral and complemented [9]. A variety V has the Fraser-Horn property if each θ ∈ Con(A1×A2), A1, A2 ∈ V , is of the form θ1 × θ2, with θi ∈ Con(Ai), i = 1, 2. This property is well known to have major implications for the structure theory of the algebras in the variety (see for example, [1], [2], [3], [7], [8], [14]). In [15] and [16] we used central elements for characterizing the varieties with the Fraser-Horn property for which the Pierce sheaf [13] (is Hausdorff and) has only directly indecomposable stalks. In this paper we will use central elements to prove the following Theorem 1. If V is a congruence modular variety such that no non-trivial algebra of V has a trivial subalgebra, then V has the Fraser-Horn property. Also, we prove a result which shows that in congruence modular varieties with ~0 and ~1 the central elements have the fundamental properties of central elements Received by the editors April 24, 1997 and, in revised form, July 7, 1997. 1991 Mathematics Subject Classification. Primary 08A05, 08B10. This research was supported by CONICOR and SECYT (UNC). 1If the language of V has a constant, then we can remove the variable w. c ©1999 American Mathematical Society