# L-group

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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.