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m (fixing superscripts)
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{{TEX|done}}
 
{{TEX|done}}
  
'' $  l $-
+
'' $  l $-group''
group''
 
  
 
A [[Group|group]]  $  G $
 
A [[Group|group]]  $  G $
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the inequality  $  a \leq  b $
 
the inequality  $  a \leq  b $
 
implies  $  xay \leq  xby $.  
 
implies  $  xay \leq  xby $.  
Similarly, a lattice-ordered group can be defined as an [[Algebraic system|algebraic system]] of signature  $  \langle  \cdot , {}  ^ {-} 1 , e, \wedge, \lor \rangle $
+
Similarly, a lattice-ordered group can be defined as an [[Algebraic system|algebraic system]] of signature  $  \langle  \cdot , {}  ^ {- 1} , e, \wedge, \lor \rangle $
that satisfies the axioms: 3)  $  \langle  G, \cdot , {}  ^ {-} 1 , e\rangle $
+
that satisfies the axioms: 3)  $  \langle  G, \cdot , {}  ^ {- 1} , e\rangle $
 
is a group; 4)  $  \langle  G, \lor , \wedge\rangle $
 
is a group; 4)  $  \langle  G, \lor , \wedge\rangle $
 
is a lattice; and 5)  $  x( y \lor z) t = xyt \lor xzt $
 
is a lattice; and 5)  $  x( y \lor z) t = xyt \lor xzt $
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The lattice of elements of a lattice-ordered group is distributive (cf. [[Distributive lattice|Distributive lattice]]). The absolute value (respectively, the positive and the negative part) of an element  $  x $
 
The lattice of elements of a lattice-ordered group is distributive (cf. [[Distributive lattice|Distributive lattice]]). The absolute value (respectively, the positive and the negative part) of an element  $  x $
is the element  $  | x | = x \lor x  ^ {-} 1 $(
+
is the element  $  | x | = x \lor x  ^ {- 1} $ (respectively,  $  x  ^ {+} = x\lor e $
respectively,  $  x  ^ {+} = x\lor e $
 
 
and  $  x  ^ {-} = x \wedge e $).  
 
and  $  x  ^ {-} = x \wedge e $).  
 
In lattice-ordered groups, the following relations hold:
 
In lattice-ordered groups, the following relations hold:
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$$  
 
$$  
 
x  =  x  ^ {+} x  ^ {-} ,\ \  
 
x  =  x  ^ {+} x  ^ {-} ,\ \  
| x |  ^ {-} 1 \leq  x  \leq  | x | ,
+
| x |  ^ {- 1}  \leq  x  \leq  | x | ,
 
$$
 
$$
  
 
$$  
 
$$  
| x |  =  x  ^ {+} ( x  ^ {-} )  ^ {-} 1 ,\  x  ^ {+} \wedge ( x  ^ {-} )  ^ {-} =  e,
+
| x |  =  x  ^ {+} ( x  ^ {-} )  ^ {- 1} ,\  x  ^ {+} \wedge ( x  ^ {-} )  ^ {- 1 } =  e,
 
$$
 
$$
  
 
$$  
 
$$  
( x \lor y)  ^ {-} 1 =  x  ^ {-} 1 \wedge y  ^ {-} 1 ,\ \  
+
( x \lor y)  ^ {- 1}  =  x  ^ {- 1} \wedge y  ^ {- 1} ,\ \  
( x \wedge y)  ^ {-} 1 =  x  ^ {-} 1 \lor y  ^ {-} 1 .
+
( x \wedge y)  ^ {- 1}  =  x  ^ {- 1} \lor y  ^ {- 1} .
 
$$
 
$$
  
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A subset  $  H $
 
A subset  $  H $
of an  $  l $-
+
of an  $  l $-group  $  G $
group  $  G $
+
is called an  $  l $-subgroup if  $  H $
is called an  $  l $-
 
subgroup if  $  H $
 
 
is a subgroup and a sublattice in  $  G $;  
 
is a subgroup and a sublattice in  $  G $;  
an  $  l $-
+
an  $  l $-subgroup  $  H $
subgroup  $  H $
+
is called an  $  l $-ideal of  $  G $
is called an  $  l $-
 
ideal of  $  G $
 
 
if it is normal and convex in  $  G $.  
 
if it is normal and convex in  $  G $.  
The set of  $  l $-
+
The set of  $  l $-subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of  $  l $-ideals of a lattice-ordered group is distributive. An  $  l $-homomorphism of an  $  l $-group  $  G $
subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of  $  l $-
+
into an  $  l $-group  $  H $
ideals of a lattice-ordered group is distributive. An  $  l $-
 
homomorphism of an  $  l $-
 
group  $  G $
 
into an  $  l $-
 
group  $  H $
 
 
is a [[Homomorphism|homomorphism]]  $  \phi $
 
is a [[Homomorphism|homomorphism]]  $  \phi $
 
of the group  $  G $
 
of the group  $  G $
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$$
 
$$
  
The kernels of  $  l $-
+
The kernels of  $  l $-homomorphisms are precisely the  $  l $-ideals of  $  l $-groups. If  $  G $
homomorphisms are precisely the  $  l $-
+
is an  $  l $-group and  $  M \subset  G $,  
ideals of  $  l $-
 
groups. If  $  G $
 
is an  $  l $-
 
group and  $  M \subset  G $,  
 
 
then the set  $  M  ^  \perp  = \{ {x \in G } : {| x | \wedge | m | = e  \textrm{ for  every  }  m \in M } \} $
 
then the set  $  M  ^  \perp  = \{ {x \in G } : {| x | \wedge | m | = e  \textrm{ for  every  }  m \in M } \} $
is a convex  $  l $-
+
is a convex  $  l $-subgroup in  $  G $ (cf. [[Convex subgroup|Convex subgroup]]).
subgroup in  $  G $(
 
cf. [[Convex subgroup|Convex subgroup]]).
 
  
 
The group  $  A( L) $
 
The group  $  A( L) $
 
of one-to-one order-preserving mappings of a totally ordered set  $  L $
 
of one-to-one order-preserving mappings of a totally ordered set  $  L $
onto itself is an  $  l $-
+
onto itself is an  $  l $-group (if for  $  f, g \in A( L) $
group (if for  $  f, g \in A( L) $
 
 
one assumes that  $  f \leq  g $
 
one assumes that  $  f \leq  g $
 
if and only if  $  f( \alpha ) \leq  g( \alpha ) $
 
if and only if  $  f( \alpha ) \leq  g( \alpha ) $
 
for all  $  \alpha \in L $).  
 
for all  $  \alpha \in L $).  
Every  $  l $-
+
Every  $  l $-group is  $  l $-isomorphic to an  $  l $-subgroup of the lattice-ordered group  $  A( L) $
group is  $  l $-
 
isomorphic to an  $  l $-
 
subgroup of the lattice-ordered group  $  A( L) $
 
 
for a suitable set  $  L $.
 
for a suitable set  $  L $.
  
The class of all lattice-ordered groups is a variety of signature  $  \langle  \cdot , {}  ^ {-} 1 , e, \wedge, \lor\rangle $(
+
The class of all lattice-ordered groups is a variety of signature  $  \langle  \cdot , {}  ^ {- 1} , e, \wedge, \lor\rangle $ (cf. [[Variety of groups|Variety of groups]]). Its most important subvariety is the class of lattice-ordered groups that can be approximated by totally ordered groups (the class of representable  $  l $-groups, cf. also [[Totally ordered group|Totally ordered group]]).
cf. [[Variety of groups|Variety of groups]]). Its most important subvariety is the class of lattice-ordered groups that can be approximated by totally ordered groups (the class of representable  $  l $-
 
groups, cf. also [[Totally ordered group|Totally ordered group]]).
 
  
 
====References====
 
====References====

Revision as of 04:03, 15 June 2022


$ l $-group

A group $ G $ on the set of elements of which a partial-order relation $ \leq $ is defined possessing the properties: 1) $ G $ is a lattice relative to $ \leq $, i.e. for any $ x, y \in G $ there are elements $ x \wedge y $, $ x \lor y $ such that $ x \wedge y \leq x, y $ and $ x \lor y \geq x, y $; for any $ z \in G $, $ z \leq x, y $ implies $ z \leq x \wedge y $, and for any $ t \in G $ and $ x, y \leq t $ one has $ x \lor y \leq t $; and 2) for any $ a, b, x, y \in G $ the inequality $ a \leq b $ implies $ xay \leq xby $. Similarly, a lattice-ordered group can be defined as an algebraic system of signature $ \langle \cdot , {} ^ {- 1} , e, \wedge, \lor \rangle $ that satisfies the axioms: 3) $ \langle G, \cdot , {} ^ {- 1} , e\rangle $ is a group; 4) $ \langle G, \lor , \wedge\rangle $ is a lattice; and 5) $ x( y \lor z) t = xyt \lor xzt $ and $ x( y \wedge z) t = xyt \wedge xzt $ for any $ x, y, z, t \in G $.

The lattice of elements of a lattice-ordered group is distributive (cf. Distributive lattice). The absolute value (respectively, the positive and the negative part) of an element $ x $ is the element $ | x | = x \lor x ^ {- 1} $ (respectively, $ x ^ {+} = x\lor e $ and $ x ^ {-} = x \wedge e $). In lattice-ordered groups, the following relations hold:

$$ x = x ^ {+} x ^ {-} ,\ \ | x | ^ {- 1} \leq x \leq | x | , $$

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

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

Two elements $ x $ and $ y $ are called orthogonal if $ | x | \lor | y | = e $. Orthogonal elements commute.

A subset $ H $ of an $ l $-group $ G $ is called an $ l $-subgroup if $ H $ is a subgroup and a sublattice in $ G $; an $ l $-subgroup $ H $ is called an $ l $-ideal of $ G $ if it is normal and convex in $ G $. The set of $ l $-subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of $ l $-ideals of a lattice-ordered group is distributive. An $ l $-homomorphism of an $ l $-group $ G $ into an $ l $-group $ H $ is a homomorphism $ \phi $ of the group $ G $ into the group $ H $ such that

$$ \phi ( x \lor y) = \phi ( x) \lor \phi ( y) ,\ \ \phi ( x \wedge y) = \phi ( x) \wedge \phi ( y). $$

The kernels of $ l $-homomorphisms are precisely the $ l $-ideals of $ l $-groups. If $ G $ is an $ l $-group and $ M \subset G $, then the set $ M ^ \perp = \{ {x \in G } : {| x | \wedge | m | = e \textrm{ for every } m \in M } \} $ is a convex $ l $-subgroup in $ G $ (cf. Convex subgroup).

The group $ A( L) $ of one-to-one order-preserving mappings of a totally ordered set $ L $ onto itself is an $ l $-group (if for $ f, g \in A( L) $ one assumes that $ f \leq g $ if and only if $ f( \alpha ) \leq g( \alpha ) $ for all $ \alpha \in L $). Every $ l $-group is $ l $-isomorphic to an $ l $-subgroup of the lattice-ordered group $ A( L) $ for a suitable set $ L $.

The class of all lattice-ordered groups is a variety of signature $ \langle \cdot , {} ^ {- 1} , e, \wedge, \lor\rangle $ (cf. Variety of groups). Its most important subvariety is the class of lattice-ordered groups that can be approximated by totally ordered groups (the class of representable $ l $-groups, cf. also Totally ordered group).

References

[1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)

Comments

References

[a1] M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988)
[a2] A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989)
[a3] J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989)
How to Cite This Entry:
Lattice-ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice-ordered_group&oldid=52434
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article