Archimedean - like Classes of Lattice - Ordered Groups
نویسنده
چکیده
Suppose <? denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) Res (£?), the class of all lattice-ordered groups which are subdirect products of groups in C; (2) Hyp(C), the class of lattice-ordered groups in Res(C) having all their ¿-homomorphic images in Res(<?); Para(C), the class of lattice-ordered groups having all their principal convex ¿-subgroups in ResíO. If C is the class of archimedean totally ordered groups then Para((?) is the class of archimedean lattice-ordered groups, Res(Ç) is the class of subdirect products of reals, and Hyp(C) consists of all the hyper archimedean lattice-ordered groups. We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex ¿-subgroup in Hyp(C); this socalled hyper-C-kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper-<?-kernels. For any class C as above we show that the free lattice-ordered group on a set X in the variety generated by C is always in Res(C). We also prove that Res ((?) has free products. Introduction. The theory presented in this paper grew out of an attempt to abstract the properties of the class of archimedean lattice-ordered groups (henceforth: /-groups), and some of its subclasses. The classes we construct offer an appetizing alternative to studying varieties of /-groups, one which quite surely has generalizations in other universal algebraic settings, although the reader may satisfy himself as to the special nature of many of the arguments. For any class C of totally ordered groups (henceforth: o-groups) which is closed under taking subgroups and quotients by o-homomorphisms, we construct Res (c), the class of /-groups which are subdirectly representable in a product of o-groups in C An /-group is hyper-<2 if it is in Res (C) and every /-homomorphic image is in Res ((2). The main result in §1 says that every representable /-group contains a unique maximal convex /-subgroup which is hyper-C; we call this the Presented to the Society, August 31, 1972; received by the editors April 3, 1972. AMS (MOS) subject classifications (1970). Primary 06A55.
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