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''lattice-ordered group''
 
''lattice-ordered group''
  
A partially ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100202.png" /> (cf. [[O-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100203.png" />-group]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100204.png" /> is a [[Lattice|lattice]] (cf. also [[Lattice-ordered group|Lattice-ordered group]]). It is useful to consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100205.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100206.png" /> as an [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100208.png" /> is a [[Group|group]] with identity element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l1100209.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002010.png" /> is a lattice with join and meet operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002011.png" /> in the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002012.png" />. The following identities hold in any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002013.png" />-group:
+
A partially ordered group $  \{ G; \cdot, \cle \} $(
 +
cf. [[O-group| $  o $-
 +
group]]) such that $  \{ G; \cle \} $
 +
is a [[Lattice|lattice]] (cf. also [[Lattice-ordered group|Lattice-ordered group]]). It is useful to consider the l $-
 +
group $  G $
 +
as an [[Algebraic system|algebraic system]] $  \{ G; \cdot,e, ^ {- 1 } , \lor, \wedge \} $,  
 +
where $  \{ G; \cdot,e, ^ {- 1 } \} $
 +
is a [[Group|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002014.png" /></td> </tr></table>
+
$$
 +
x ( y \lor z ) t = xyt \lor xzt,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002015.png" /></td> </tr></table>
+
$$
 +
x ( y \wedge z ) t = xyt \wedge xzt.
 +
$$
  
The lattice of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002016.png" />-group is distributive (cf. [[Distributive lattice|Distributive lattice]]). The class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002017.png" />-groups is a variety of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002018.png" /> (cf. [[L-variety|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002019.png" />-variety]]); it is locally closed, and closed under taking direct and Cartesian products, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002021.png" />-subgroups (i.e., subgroups that are sublattices), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002023.png" />-homomorphisms (i.e., homomorphisms that preserve the group operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002024.png" /> and the lattice operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002025.png" />).
+
The lattice of an l $-
 +
group is distributive (cf. [[Distributive lattice|Distributive lattice]]). The class of all l $-
 +
groups is a variety of signature $  \{ \cdot,e, ^ {- 1 } , \lor, \wedge \} $(
 +
cf. [[L-variety| 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002027.png" />-groups are: 1) the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002028.png" /> of the set of real-valued continuous functions defined on the real number set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002029.png" />, with the order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002030.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002031.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002033.png" />; and 2) the automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002034.png" /> of a totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002035.png" /> with order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002036.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002037.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002038.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002039.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002040.png" />-groups is used in the study of the structure of ordered vector spaces, function spaces and infinite groups, in particular for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002041.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002042.png" />-groups is that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002043.png" />-group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002044.png" />-isomorphic to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002045.png" />-subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002046.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002047.png" /> for a suitable totally ordered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002048.png" />. Using this theorem, it can be proved that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002049.png" />-group is imbeddable in a divisible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002050.png" />-group as well as in a simple group. The class of groups that may be endowed with the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002051.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002052.png" />-group.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002053.png" />-group is a torsion-free group and has a decomposition property: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002055.png" /> for positive elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002058.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002059.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002060.png" />-group and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002063.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002064.png" />. Then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002065.png" /></td> </tr></table>
+
$$
 +
x = x  ^ {+} x  ^ {-} ,  x  ^ {+} \wedge ( x  ^ {-} ) ^ {- 1 } = e,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002066.png" /></td> </tr></table>
+
$$
 +
\left | x \right | = x  ^ {+} ( x  ^ {-} ) ^ {- 1 } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002067.png" /></td> </tr></table>
+
$$
 +
\left | {x \lor y } \right | \cle \left | x \right | \lor \left | y \right | \cle \left | x \right | \left | y \right | ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002068.png" /></td> </tr></table>
+
$$
 +
\left | {xy } \right | \cle \left | x \right | \left | y \right | \left | x \right | ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002069.png" /></td> </tr></table>
+
$$
 +
( x \lor y ) ^ {- 1 } = x ^ {- 1 } \wedge y ^ {- 1 } .
 +
$$
  
Elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002070.png" /> are called orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002072.png" />. Orthogonal elements commute.
+
Elements $  x,y \in G $
 +
are called orthogonal if $  | x | \wedge | y | = e $.  
 +
Orthogonal elements commute.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002073.png" />-group may be described by its positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002075.png" />, for which the following properties hold:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002076.png" />;
+
1) $  P \cdot P \subseteq P $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002077.png" />;
+
2) $  P \cap P = \{ e \} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002078.png" />;
+
3) $  \forall x: x ^ {- 1 } Px \subseteq P $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002079.png" /> is a lattice respect with the partial order induced from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002080.png" />. If, in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002081.png" />, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002082.png" /> with the properties 1)–4) can be found, then it is possible to turn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002083.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002084.png" />-group by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002085.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002086.png" />. It is correct to identify the order in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002087.png" />-group with its positive cone. The notation  "l-group"  is connected with the notation for right-ordered groups (cf. [[Ro-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002088.png" />-group]]). In particular, the positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002089.png" /> of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002090.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002091.png" /> is the intersection of a suitable set of right orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002092.png" /> on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002093.png" />.
+
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| $  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002094.png" />-group in terms of convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002095.png" />-subgroups (cf. [[Convex subgroup|Convex subgroup]]). A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002096.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002097.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l11002098.png" /> is called a convex subgroup if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020101.png" />:
+
It is useful to describe the structure of an l $-
 +
group in terms of convex l $-
 +
subgroups (cf. [[Convex subgroup|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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020102.png" /></td> </tr></table>
+
$$
 +
x \cle z \cle y \Rightarrow z \in H.
 +
$$
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020103.png" /> of all convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020104.png" />-subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020105.png" /> is a complete sublattice of the lattice of all subgroups (cf. [[Complete lattice|Complete lattice]]). A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020106.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020107.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020108.png" /> is the kernel of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020109.png" />-homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020110.png" /> if and only if it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020113.png" />-ideal, i.e., a normal convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020114.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020115.png" />.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020116.png" /> is a subset of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020117.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020118.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020119.png" /> is called a polar. Every polar in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020121.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020122.png" /> is a convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020123.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020124.png" />. The following properties hold for polars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020126.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020127.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020128.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020129.png" /></td> </tr></table>
+
$$
 +
M ^ {\perp  \perp  \perp  } = M  ^  \perp  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020130.png" /></td> </tr></table>
+
$$
 +
M \subseteq N \Rightarrow M  ^  \perp  \supseteq N  ^  \perp  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020131.png" /></td> </tr></table>
+
$$
 +
M  ^  \perp  \cap N  ^  \perp  = ( M \cup N )  ^  \perp  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020132.png" /></td> </tr></table>
+
$$
 +
( M  ^  \perp  \cup N  ^  \perp  )  ^  \perp  = M ^ {\perp  \perp  } \cap N ^ {\perp  \perp  } .
 +
$$
  
The set of all polars of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020133.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020134.png" /> is a [[Boolean algebra|Boolean algebra]], but not a sublattice of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020135.png" />. The properties and the significance of polars are well investigated.
+
The set of all polars of an l $-
 +
group $  G $
 +
is a [[Boolean algebra|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|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020136.png" />-group]] is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020137.png" />-group with a total order (cf. also [[Totally ordered group|Totally ordered group]]). If an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020138.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020139.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020140.png" />-subgroup of the Cartesian product of totally ordered groups, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020141.png" /> is called a representable group. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020142.png" /> of representable groups has been well investigated. It is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020143.png" />-variety given by the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020144.png" /> in the variety of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020145.png" />-groups. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020146.png" />-group is representable if and only if every polar of it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020147.png" />-ideal. The positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020148.png" /> of a representable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020149.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020150.png" /> is the intersection of all total orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020151.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020152.png" />. Every locally nilpotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020153.png" />-group is representable.
+
An [[O-group| $  o $-
 +
group]] is an l $-
 +
group with a total order (cf. also [[Totally ordered group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020154.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020155.png" /> is called Archimedean if the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020157.png" /> holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020158.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020159.png" /> for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020160.png" />. Every Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020161.png" />-group is Abelian (cf. [[Abelian group|Abelian group]]) and it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020162.png" />-subgroup of the Cartesian product of copies of the totally ordered additive group of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020163.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020164.png" /> of Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020165.png" />-groups is closed under formation of subgroups, direct and Cartesian products. It is not closed under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020166.png" />-homomorphisms and is not an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020167.png" />-variety. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020168.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020169.png" /> of real-valued functions on a compact topologic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110020/l110020170.png" /> is Archimedean.
+
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|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|Lattice-ordered group]] (Volume 5).
 
This article extends and complements the article [[Lattice-ordered group|Lattice-ordered group]] (Volume 5).

Latest revision as of 22:15, 5 June 2020


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)
How to Cite This Entry:
L-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-group&oldid=17768
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article