L-group

lattice-ordered group

A partially ordered group $ \{ G; \cdot, \cle \} $( cf. $ o $- group) such that $ \{ G; \cle \} $ is a lattice (cf. also Lattice-ordered group). It is useful to consider the $ l $- group $ G $ as an algebraic system $ \{ G; \cdot,e, ^ {- 1 } , \lor, \wedge \} $, where $ \{ G; \cdot,e, ^ {- 1 } \} $ is a group with identity element $ e $, and $ \{ G; \lor, \wedge \} $ is a lattice with join and meet operations $ \lor, \wedge $ in the lattice $ \{ G; \cle \} $. The following identities hold in any $ l $- group:

$$ x ( y \lor z ) t = xyt \lor xzt, $$

$$ x ( y \wedge z ) t = xyt \wedge xzt. $$

The lattice of an $ l $- group is distributive (cf. Distributive lattice). The class of all $ l $- groups is a variety of signature $ \{ \cdot,e, ^ {- 1 } , \lor, \wedge \} $( cf. $ l $- variety); it is locally closed, and closed under taking direct and Cartesian products, $ l $- subgroups (i.e., subgroups that are sublattices), and $ l $- homomorphisms (i.e., homomorphisms that preserve the group operation $ \cdot $ and the lattice operations $ \lor, \wedge $).

The most important examples of $ l $- groups are: 1) the additive group $ C [ \mathbf R ] $ of the set of real-valued continuous functions defined on the real number set $ \mathbf R $, with the order: $ f \cle g $, for $ f,g \in C [ \mathbf R ] $, if and only if $ f ( x ) \cle g ( x ) $ for all $ x \in \mathbf R $; and 2) the automorphism group $ { \mathop{\rm Aut} } ( X ) $ of a totally ordered set $ X $ with order: $ \varphi \cle \psi $, for $ \varphi, \psi \in { \mathop{\rm Aut} } ( X ) $, if and only if $ x \varphi \cle x \psi $ for all $ x \in X $.

The theory of $ l $- groups is used in the study of the structure of ordered vector spaces, function spaces and infinite groups, in particular for the groups $ { \mathop{\rm Aut} } ( X ) $.

The most important fact of the theory of $ l $- groups is that every $ l $- group is $ l $- isomorphic to some $ l $- subgroup of the $ l $- group $ { \mathop{\rm Aut} } ( X ) $ for a suitable totally ordered $ X $. Using this theorem, it can be proved that every $ l $- group is imbeddable in a divisible $ l $- group as well as in a simple group. The class of groups that may be endowed with the structure of an $ l $- 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 $ l $- group.

Every $ l $- group is a torsion-free group and has a decomposition property: if $ a \cle b _ {1} \dots b _ {n} $ for positive elements $ a,b _ {1} \dots b _ {n} $, then $ a = c _ {1} \dots c _ {n} $, where $ e \cle c _ {i} \cle b _ {i} $.

Let $ G $ be an $ l $- group and put $ x ^ {+} = x \lor e $, $ x ^ {-} = x \wedge e $, $ | x | = x \lor x ^ {- 1 } $ for $ x \in G $. Then

$$ x = x ^ {+} x ^ {-} , x ^ {+} \wedge ( x ^ {-} ) ^ {- 1 } = e, $$

$$ \left | x \right | = x ^ {+} ( x ^ {-} ) ^ {- 1 } , $$

$$ \left | {x \lor y } \right | \cle \left | x \right | \lor \left | y \right | \cle \left | x \right | \left | y \right | , $$

$$ \left | {xy } \right | \cle \left | x \right | \left | y \right | \left | x \right | , $$

$$ ( x \lor y ) ^ {- 1 } = x ^ {- 1 } \wedge y ^ {- 1 } . $$

Elements $ x,y \in G $ are called orthogonal if $ | x | \wedge | y | = e $. Orthogonal elements commute.

An $ l $- group may be described by its positive cone $ P = P ( G ) = \{ {x \in G } : {x \cge e } \} $, for which the following properties hold:

1) $ P \cdot P \subseteq P $;

2) $ P \cap P = \{ e \} $;

3) $ \forall x: x ^ {- 1 } Px \subseteq P $;

4) $ P $ is a lattice respect with the partial order induced from $ G $. If, in a group $ G $, a set $ P $ with the properties 1)–4) can be found, then it is possible to turn $ G $ in an $ l $- group by setting $ x \cle y $ if and only if $ yx ^ {- 1 } \in P $. It is correct to identify the order in an $ l $- group with its positive cone. The notation "l-group" is connected with the notation for right-ordered groups (cf. $ ro $- group). In particular, the positive cone $ P ( G ) $ of any $ l $- group $ G $ is the intersection of a suitable set of right orders $ P _ \alpha $ on the group $ G $.

It is useful to describe the structure of an $ l $- group in terms of convex $ l $- subgroups (cf. Convex subgroup). A subgroup $ H $ of an $ l $- group $ G $ is called a convex subgroup if for all $ x,y \in H $, $ z \in G $:

$$ x \cle z \cle y \Rightarrow z \in H. $$

The set $ {\mathcal C} ( G ) $ of all convex $ l $- subgroups of $ G $ is a complete sublattice of the lattice of all subgroups (cf. Complete lattice). A subset $ N $ of an $ l $- group $ G $ is the kernel of an $ l $- homomorphism of $ G $ if and only if it is an $ l $- ideal, i.e., a normal convex $ l $- subgroup of $ G $.

If $ M $ is a subset of an $ l $- group $ G $, then the set $ M ^ \perp = \{ {x \in G } : {| x | \wedge | m | = e \textrm{ for all } m \in M } \} $ is called a polar. Every polar in a $ l $- group $ G $ is a convex $ l $- subgroup of $ G $. The following properties hold for polars $ M $ and $ N $ of an $ l $- group $ G $:

$$ M ^ {\perp \perp \perp } = M ^ \perp , $$

$$ M \subseteq N \Rightarrow M ^ \perp \supseteq N ^ \perp , $$

$$ M ^ \perp \cap N ^ \perp = ( M \cup N ) ^ \perp , $$

$$ ( M ^ \perp \cup N ^ \perp ) ^ \perp = M ^ {\perp \perp } \cap N ^ {\perp \perp } . $$

The set of all polars of an $ l $- group $ G $ is a Boolean algebra, but not a sublattice of the lattice $ {\mathcal C} ( G ) $. The properties and the significance of polars are well investigated.

An $ o $- group is an $ l $- group with a total order (cf. also Totally ordered group). If an $ l $- group $ G $ is an $ l $- subgroup of the Cartesian product of totally ordered groups, then $ G $ is called a representable group. The class $ {\mathcal R} $ of representable groups has been well investigated. It is the $ l $- variety given by the identity $ ( x \wedge y ^ {- 1 } x ^ {- 1 } y ) \lor e = e $ in the variety of all $ l $- groups. An $ l $- group is representable if and only if every polar of it is an $ l $- ideal. The positive cone $ P $ of a representable $ l $- group $ G $ is the intersection of all total orders of $ G $ restricted to $ P $. Every locally nilpotent $ l $- group is representable.

An $ l $- group $ G $ is called Archimedean if the equality $ b = e $ holds for all $ a,b \in G $ such that $ a ^ {n} \leq b $ for any integer $ n $. Every Archimedean $ l $- group is Abelian (cf. Abelian group) and it is an $ l $- subgroup of the Cartesian product of copies of the totally ordered additive group of real numbers $ \mathbf R $. The class $ {\mathcal A} $ of Archimedean $ l $- groups is closed under formation of subgroups, direct and Cartesian products. It is not closed under $ l $- homomorphisms and is not an $ l $- variety. The $ l $- group $ C [ X, \mathbf R ] $ of real-valued functions on a compact topologic space $ X $ is Archimedean.

This article extends and complements the article Lattice-ordered group (Volume 5).

References

[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)