Is the Lattice of Torsion Classes Algebraic?

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not. As a consequence of this, one obtains a (local) decomposition theorem which is canonical (in every sense of the word) and always works. The general terminology and notation of this paper is standard. It is perhaps worthwhile to make the following observations: l-group is the abbreviation for lattice-ordered group. In an /-group G, a value is a convex /-subgroup M, maximal with respect to the property of missing some element x G G. M is also said to be a value of x. For a reference concerning the general theory of /-groups, the reader may consult [1]. The theory of torsion classes used here is not deep. In any case, we refer the reader to [4] and [5]. In addition, we review the main definitions. A torsion class is a class of /-groups closed under taking (1) convex /-subgroups, (2) /homomorphic images and (3) forming joins of convex /-subgroups already in the class. If 5" is a torsion class and G is an /-group, 5"(G ) stands for the join of all the convex /-subgroups of G in ?T. 'öiG ) is a characteristic /-ideal, and the largest convex /-subgroup of G in ?T. It is called the torsion radical of G with respect to 5". If 5" is a torsion class we get [4, Proposition 1.1]: (a) for each convex /subgroup K of G, 5{K) = K n 5(G), and (b) if : G -» H is an /-epimorphism, [5(G)]$ Q 5(H). Conversely, a function subject to (a) and (b) above gives rise to a torsion class by setting 5 = {G\5(G) = G); this correspondence between classes and radicals is one-to-one [1, Proposition 1.2]. The lattice of torsion classes-itself a proper class-is a Brouwer lattice. The meet operation is class intersection, while the join is given by: 5 = V/e/3) if and only if 5(G) = 2,e/ ^/(G), for each /-group G. In this note we will show this lattice to be completely distributive. Received by the editors March 26, 1976. AMS (MOS) subject classifications (1970). Primary 06A35, 06A60; Secondary 18E40.