A partially ordered group (cf. -group) such that is a lattice (cf. also Lattice-ordered group). It is useful to consider the -group as an algebraic system , where is a group with identity element , and is a lattice with join and meet operations in the lattice . The following identities hold in any -group:
The lattice of an -group is distributive (cf. Distributive lattice). The class of all -groups is a variety of signature (cf. -variety); it is locally closed, and closed under taking direct and Cartesian products, -subgroups (i.e., subgroups that are sublattices), and -homomorphisms (i.e., homomorphisms that preserve the group operation and the lattice operations ).
The most important examples of -groups are: 1) the additive group of the set of real-valued continuous functions defined on the real number set , with the order: , for , if and only if for all ; and 2) the automorphism group of a totally ordered set with order: , for , if and only if for all .
The theory of -groups is used in the study of the structure of ordered vector spaces, function spaces and infinite groups, in particular for the groups .
The most important fact of the theory of -groups is that every -group is -isomorphic to some -subgroup of the -group for a suitable totally ordered . Using this theorem, it can be proved that every -group is imbeddable in a divisible -group as well as in a simple group. The class of groups that may be endowed with the structure of an -group is large. E.g., it contains the classes of Abelian torsion-free groups, locally nilpotent torsion-free groups, and many others. There are torsion-free groups that cannot be imbedded in any -group.
Every -group is a torsion-free group and has a decomposition property: if for positive elements , then , where .
Let be an -group and put , , for . Then
Elements are called orthogonal if . Orthogonal elements commute.
An -group may be described by its positive cone , for which the following properties hold:
4) is a lattice respect with the partial order induced from . If, in a group , a set with the properties 1)–4) can be found, then it is possible to turn in an -group by setting if and only if . It is correct to identify the order in an -group with its positive cone. The notation "l-group" is connected with the notation for right-ordered groups (cf. -group). In particular, the positive cone of any -group is the intersection of a suitable set of right orders on the group .
It is useful to describe the structure of an -group in terms of convex -subgroups (cf. Convex subgroup). A subgroup of an -group is called a convex subgroup if for all , :
The set of all convex -subgroups of is a complete sublattice of the lattice of all subgroups (cf. Complete lattice). A subset of an -group is the kernel of an -homomorphism of if and only if it is an -ideal, i.e., a normal convex -subgroup of .
If is a subset of an -group , then the set is called a polar. Every polar in a -group is a convex -subgroup of . The following properties hold for polars and of an -group :
The set of all polars of an -group is a Boolean algebra, but not a sublattice of the lattice . The properties and the significance of polars are well investigated.
An -group is an -group with a total order (cf. also Totally ordered group). If an -group is an -subgroup of the Cartesian product of totally ordered groups, then is called a representable group. The class of representable groups has been well investigated. It is the -variety given by the identity in the variety of all -groups. An -group is representable if and only if every polar of it is an -ideal. The positive cone of a representable -group is the intersection of all total orders of restricted to . Every locally nilpotent -group is representable.
An -group is called Archimedean if the equality holds for all such that for any integer . Every Archimedean -group is Abelian (cf. Abelian group) and it is an -subgroup of the Cartesian product of copies of the totally ordered additive group of real numbers . The class of Archimedean -groups is closed under formation of subgroups, direct and Cartesian products. It is not closed under -homomorphisms and is not an -variety. The -group of real-valued functions on a compact topologic space is Archimedean.
This article extends and complements the article Lattice-ordered group (Volume 5).
|[a1]||L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)|
|[a2]||A. Bigard, K. Keimel, S. Wolfenstein, "Groupes et anneaux rétiqulés" , Springer (1977)|
|[a3]||"Lattice-ordered groups: advances and techniques" A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Kluwer Acad. Publ. (1989)|
|[a4]||V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)|
L-group. V.M. Kopytov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=L-group&oldid=17768