The distributive lattice free product as a sublattice of the abelian<i>l</i>-group free product

Powell, Wayne B.; Tsinakis, Constantine

doi:10.1017/s1446788700019789

Cited by 8 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One of the more important classes of ordered systems is the variety of vector lattices. It is easy to adjust the proofs of Theorems 1 and 2 to get analogous results for vector lattices (see Bernau [2] and the authors' paper [16]), and in so doing achieve the following theorem.…”

Section: Henceforth We Let G/ = 4>(g¡) = ^(G)mentioning

confidence: 94%

See 1 more Smart Citation

Free products of abelian 𝑙-groups are cardinally indecomposable

Powell¹,

Tsinakis²

1982

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Henceforth We Let G/ = 4>(g¡) = ^(G)mentioning

confidence: 94%

“…Inasmuch as any free abelian /-group is a free product of copies of Z ffl Z, our result extends the latter result. Our proof utilizes one of Bernau's aforementioned results, and a construction of abelian /-group free products in terms of free extensions of partially ordered groups as was presented by the authors in [16] (see Theorem 2 below).…”

mentioning

confidence: 99%

Free products of abelian 𝑙-groups are cardinally indecomposable

Powell¹,

Tsinakis²

1982

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us begin by first mentioning a result which is already complete. Let 6 ύ e be the class of distributive lattices with a distinguished element e. Every abelian /-group can be viewed as a member of 6 ϋ e with 0 = e. Using Theorem 2.6 we show in a forthcoming paper [27] that the ^D e -free product of a family (G ι ,\ i G ί) of abelian /-groups is the sublattice of the abelian /-group free product generated by U.^G,. The corresponding result for the class £ of all /-groups has been established by Franchello in [10].…”

Section: Further Applications and Open Problemsmentioning

confidence: 98%

Free products in the class of abelianl-groups

Powell¹,

Tsinakis²

1983

Pacific J. Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Universal Aspects of the Theory of Lattice Ordered Groups

Powell

1989

Ordered Algebraic Structures

View full text Add to dashboard Cite

The distributive lattice free product as a sublattice of the abelianl-group free product

Cited by 8 publications

References 11 publications

Free products of abelian 𝑙-groups are cardinally indecomposable

Free products of abelian 𝑙-groups are cardinally indecomposable

Free products in the class of abelianl-groups

Universal Aspects of the Theory of Lattice Ordered Groups

Contact Info

Product

Resources

About