Essential adjunction of a strong unit to an archimedean lattice-ordered group

2020 ◽  
Vol 81 (3) ◽  
Author(s):  
Anthony W. Hager ◽  
Philip Scowcroft
Keyword(s):  
Lattice Ordered Group ◽  
Ordered Group ◽  
Strong Unit ◽  
Archimedean Lattice

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

On a relative uniform completion of an archimedean lattice ordered group

2009 ◽  
Vol 59 (2) ◽  
Author(s):  
Štefan Černák ◽  
Judita Lihová
Keyword(s):  
Uniform Convergence ◽  
Subdirect Product ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice ◽  
Finite Direct Product ◽  
Relatively Uniform Convergence

AbstractThe notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups.Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) $$ G_{\omega _1 } $$ of G is dealt with. It is known that $$ G_{\omega _1 } $$ exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in $$ G_{\omega _1 } $$. Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.

On the epicomplete monoreflection of an archimedean lattice-ordered group

2005 ◽  
Vol 54 (4) ◽  
pp. 417-434 ◽  
Author(s):  
R. N. Ball ◽  
A. W. Hager ◽  
D. G. Johnson ◽  
A. Kizanis
Keyword(s):  
Lattice Ordered Group ◽  
Ordered Group ◽  
Archimedean Lattice

Splitting properties in Archimedean l-groups

1977 ◽  
Vol 23 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Marlow Anderson ◽  
Paul Conrad ◽  
Otis Kenny
Keyword(s):  
Essential Extension ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Archimedean Lattice

Throughout this paper an l-group will always mean an archimedean lattice-ordered group and we shall confine our attention to such groups. An l-group splits if it is a cardinal summand of each l-group that contains it as an l-ideal. Suppose that G is an l-subgroup of an l-group H. Then G is large in H or H is an essential extension of G if for each l-ideal L≠0 of H, L∩G≠0. G is essentially closed if it does not admit any proper essential extension. Conrad (1971) proved that each essentially closed l-group splits, but not conversely.

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