Free Products in Varieties of Lattice-Ordered Groups

1989 ◽  
pp. 278-307 ◽  
Author(s):  
Wayne B. Powell ◽  
Constantine Tsinakis
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Free products of lattice ordered groups

1972 ◽  
Vol 2 (1) ◽  
pp. 247-254 ◽  
Author(s):  
C. Holland ◽  
E. Scrimger
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups

scholarly journals Right orders and amalgamation for lattice-ordered groups

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
V. Bludov ◽  
A. Glass
Keyword(s):  
Free Product ◽  
Sufficient Conditions ◽  
State University ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Amalgamated Subgroup ◽  
Necessary And Sufficient

AbstractLet H i be a sublattice subgroup of a lattice-ordered group G i (i = 1, 2). Suppose that H 1 and H 2 are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G 1 and G 2 can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H 1 and H 1φ. In this article we prove that the group free product of G 1 and G 2 amalgamating H 1 and H 1φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (ℓ-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].

Free products of lattice ordered groups

1984 ◽  
Vol 18 (2) ◽  
pp. 178-198 ◽  
Author(s):  
Wayne B. Powell ◽  
Constantine Tsinakis
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups

scholarly journals Sets of disjoint elements in free products of lattice-ordered groups

1988 ◽  
Vol 104 (4) ◽  
pp. 1014-1014 ◽  
Author(s):  
Wayne B. Powell ◽  
Constantine Tsinakis
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Sublattices of free products of lattice ordered groups

1978 ◽  
Vol 8 (1) ◽  
pp. 101-110 ◽  
Author(s):  
James D. Franchello
Keyword(s):  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups

scholarly journals On the Toeplitz algebras of right-angled and finite-type Artin groups

2002 ◽  
Vol 72 (2) ◽  
pp. 223-246 ◽  
Author(s):  
John Crisp ◽  
Marcelo Laca
Keyword(s):  
Uniqueness Theorem ◽  
Finite Type ◽  
Analogous Result ◽  
Universal Property ◽  
Artin Groups ◽  
Graph Product ◽  
Free Products ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Right Angled Artin Groups

AbstractThe graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields a uniqueness theorem for the C*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.

Generalized Commutativity of Lattice-Ordered Groups

2015 ◽  
Vol 65 (2) ◽  
Author(s):  
M. R. Darnel ◽  
W. C. Holland ◽  
H. Pajoohesh
Keyword(s):  
Theorem Proving ◽  
Natural Generalization ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Weak Commutativity

AbstractIn this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

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