Generating varieties of lattice-ordered groups: Approximating wreath products
A. M. W. Glass · Illinois Journal of Mathematics · 1986
In fond memory of Bill Boone 0. Introduction In this note we will be concerned with varieties of lattice-ordered groups, finitely presented lattice-ordered groups and wreath products of lattice-ordered groups.The totally ordered group of integers Z is finitely presented as a lattice- ordered group: Z (x; x A 1 1), where 1 denotes the group identity.Moreover, the variety 92 of Abelian lattice-ordered groups is the smallest variety of lattice-ordered groups containing Z [14].So certain "natural" non-trivial varieties of lattice-ordered groups are generated by a single finitely presented lattice-ordered group.Note that "finitely presented" means in the variety of a// lattice-ordered groups, not in the subvariety being considered.Now if G and H are finitely presented lattice-ordered groups and generate varieties 1I and 3 respectively, then clearly G m H generates 1I v 3, where G t H is the ordered direct product of G and H where (g, h) > 1 if and only if g > 1 (in G) and h > 1 (in H).Further, G m H is finitely presented: take as generators the disjoint union of the generating sets { g: I} of G and { h j: j J } of H, and as defining relations the union of those of G and H together with Ig;I ^Ihyl-1 (i I, j J), where Ixl--x /x -x.Hence the set of varieties of lattice-ordered groups generated by single finitely presented lattice-ordered groups is a join semilattice of the lattice of varieties of lattice-ordered groups.If 1I and are each generated by a single finitely presented lattice-ordered group, what about 1I N 3 and 1I3?In the case that 3 9 which is generated by Z, the answer is yes.By [14], 1I N 9A 92 except when 1I is the variety defined by x'qy (x y).The main result in this paper is therefore: