The Theory of Lattice-Ordered Groups pp 161-185 | Cite as
Lattice properties in lattice-ordered groups
Chapter
Abstract
The standard definition of the completeness of partially ordered sets is not usefull for po-groups. For example, let (a) be an infinite cyclic subgroup of po-group G generated by any positive element a. Then (a) cannot have the supremum (the least upper bound) inz G. Therefore we modify this definition as follows: a partially ordered group G is complete (or order complete) if every non-empty bounded above subset M has a supremum
$$M = \vee M = \mathop \vee \limits_{x \notin M} \{ x\} $$
Keywords
Boolean Algebra Finite Subset Positive Element Compact Element Order Completeness
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information
© Springer Science+Business Media Dordrecht 1994