Free products in the class of abelianl-groups

Abstract

The main objective of this paper is to present several constructions of free products in the class of abelian /-groups which are sufficiently concrete to allow for an in depth examination of their structure. Some applications of these constructions are discussed, and it is shown that abelian /-group free products satisfy the subalgebra property. Further, some questions on free /-groups over group free products are considered for a variety of /-groups which is either abelian or contains the representable /-groups. Finally, a general observation is made about countable chains and countable disjoint sets in free algebras.1. Introduction. Let % be a class of /-groups (lattice ordered groups) and (G ι ,\ i G ί) a family of members of %. The %-free product of this family is an /-group G G %, denoted by % |J/e3 Gi> together with a family of /-monomorphisms (a ι ,:generates G as an /-group; (ii) for every H G % and every family of /-homomorphisms (β,: G, -> i/| / G ί), there exists a (necessarily) unique /-homomorphism β: G -> H satisfying β t = βa t for all / G ί. Following the usual practice we shall speak of ^LJ/es^ as the fyUfree product of (G, | i ε ί). To simplify our notation, we use the "internal" definition of a %-free product, that is, we identify each free factor G, with its image a^G;) in ^\J iG jG i9 and thus we think of each G, as an /-subgroup of ^UieίGj As a consequence of general existence theorems (See Gratzer [13, p. 186] or Pierce [25, p. 107]), tyUfree products always exist in any class of /-groups closed under products and /-subgroups.In this paper we concentrate on the class & of abelian /-groups, although many of our results also hold in the important class of vector lattices. Our main goal is to develop a reasonable representation theory for έE-free products. This is done in §2 where we give several methods of constructing these products, among the most useful of which represents β Uieί^ί (G/ e ί£) as a subdirect product of totally ordered abelian groups each determined by the primes of the individual G/s. We also show here how the Φ free products relate to the free abelian /-groups over partially ordered abelian groups.The third and fourth sections of the paper are devoted to considering several different properties for free products of /-groups. In particular using the representation theory established in §2 we show that the subalgebra property is satisfied for έMree products.

Amalgamation and interpolation in ordered algebras

“…We also obtain a new proof of Weinberg's theorem [70] that A is generated as a quasivariety by the abelian -group Z of the integers. Amalgamation was first established for A by Pierce in [61]; other algebraic proofs are given by Powell and Tsinakis in [63,64]. The self-contained proof given below is closest to the model-theoretic approach of Weispfenning [71] based on quantifier elimination for totally and densely ordered abelian -groups.…”

“…Publications of particular relevance to our discussion include Bacsich [4], Czelakowski and Pigozzi [14], Gabbay and Maksimova [22], Galatos and Ono [24], Kihara and Ono [43,44], Madarasz [45], Maksimova [46][47][48], Montagna [53], Pierce [61], Pigozzi [62], Powell and Tsinakis [63][64][65][66], and Wroński [73,74]. We defer more precise historical and bibliographical details to the appropriate points in the text.…”

Vector lattices over subfields of the reals

“…PROOF. F is an /-subgroup of G x and G 2 so F u F is an /-subgroup of G x u G 2 [12]. Since F u F is not archimedean neither is G x u G 2 .…”

“…(4) If {G t \i E 1} is a set of l-subgroups of an abelian l-group G and each G is an F-vector lattice, then so is the I-subgroup of G that is generated by PROOF. F is an /-subgroup of G x and G 2 so F u F is an /-subgroup of G x u G 2 [12]. Since F u F is not archimedean neither is G x u G 2 .…”

Strong amalgamations of lattice ordered groups and modules

“…The proof of the next theorem will draw on representations of free products in A and M. We describe briefly this process here and refer the reader to Powell and Tsinakis [11] and Cherri and Powell [3] …”