Free products of abelian 𝑙-groups are cardinally indecomposable

Powell, Wayne B.; Tsinakis, Constantine

doi:10.1090/s0002-9939-1982-0671199-6

“…Whenever Go and Hq are (finitely generated) sublattice subgroups of the latticeordered groups G and H respectively, is the sublattice subgroup of the free product (in the category of lattice-ordered groups) of G and H generated by Go U i/o isomorphic to the free product (in this category) of Go and HqI Powell and Tsinakis [12] proved the direct indecomposability of free products of nontrivial Abelian lattice-ordered groups (in the category of Abelian lattice-ordered groups) and of nontrivial finitely generated lattice-ordered groups belonging to any proper subvariety of lattice-ordered groups (the free product being taken in the variety) [13, Corollary 8.7], but leave open the general problem for free products in the variety of all lattice-ordered groups [13, Problem 10.9]. Corollary 1' therefore answers their question for the variety of all lattice-ordered groups.…”

Section: Introductionmentioning

confidence: 99%

Free Products of Lattice-Ordered Groups

Glass¹

1987

Proceedings of the American Mathematical Society

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ABSTRACT. We give a highly homogeneous representation for the free product of two nontrivial countable lattice-ordered groups and obtain, as a consequence of the method, that the free product of nontrivial lattice-ordered groups is directly indecomposable and has trivial center.1. Introduction. The free lattice-ordered group on a countably infinite set of generators was shown to be isomorphic to a doubly transitive sublattice subgroup of the lattice-ordered group of all order-preserving permutations of the rational line (see [1 or 2, Theorem 6.7, and 9]). The proof is easily extended to give a faithful doubly transitive representation for the free lattice-ordered group on any infinite set of generators. More recently, Kopytov [6] and, independently, McCleary [9] proved by a far more subtle technique that the free lattice-ordered group on at least two generators possesses a faithful doubly transitive representation (on the rational line if the number of generators is finite or countable). The ideas were further extended in [3] to prove:1. If G is any lattice-ordered group and F is a free lattice-ordered group on an infinite set of generators of cardinality at least \G\, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on some ordered field of cardinality \F\.2. If G is any countable lattice-ordered group and F is a free lattice-ordered group on a finite number (> 1) of generators, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on the rational line.Although the proof of 2 is the more delicate, the purpose of this note is to extend it. (In contrast, I have been unable to extend the proof of 1.) Specifically,

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The Ar-Free Products Of Archimedean l-Groups

Ton¹

1998

Czechoslovak Mathematical Journal

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Free products of lattice-ordered groups

Glass¹

1987

Proc. Amer. Math. Soc.

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ABSTRACT. We give a highly homogeneous representation for the free product of two nontrivial countable lattice-ordered groups and obtain, as a consequence of the method, that the free product of nontrivial lattice-ordered groups is directly indecomposable and has trivial center.1. Introduction. The free lattice-ordered group on a countably infinite set of generators was shown to be isomorphic to a doubly transitive sublattice subgroup of the lattice-ordered group of all order-preserving permutations of the rational line (see [1 or 2, Theorem 6.7, and 9]). The proof is easily extended to give a faithful doubly transitive representation for the free lattice-ordered group on any infinite set of generators. More recently, Kopytov [6] and, independently, McCleary [9] proved by a far more subtle technique that the free lattice-ordered group on at least two generators possesses a faithful doubly transitive representation (on the rational line if the number of generators is finite or countable). The ideas were further extended in [3] to prove:1. If G is any lattice-ordered group and F is a free lattice-ordered group on an infinite set of generators of cardinality at least \G\, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on some ordered field of cardinality \F\.2. If G is any countable lattice-ordered group and F is a free lattice-ordered group on a finite number (> 1) of generators, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on the rational line.Although the proof of 2 is the more delicate, the purpose of this note is to extend it. (In contrast, I have been unable to extend the proof of 1.) Specifically,

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Free products of abelian 𝑙-groups are cardinally indecomposable

Cited by 3 publications

References 16 publications

Free Products of Lattice-Ordered Groups

Free Products of Lattice-Ordered Groups

The Ar-Free Products Of Archimedean l-Groups

Free products of lattice-ordered groups

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