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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576702.png" />-group''
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A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576703.png" /> on the set of elements of which a partial-order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576704.png" /> is defined possessing the properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576705.png" /> is a [[Lattice|lattice]] relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576706.png" />, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576707.png" /> there are elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l0576709.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767011.png" />; for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767014.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767016.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767017.png" />; and 2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767018.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767019.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767020.png" />. Similarly, a lattice-ordered group can be defined as an [[Algebraic system|algebraic system]] of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767021.png" /> that satisfies the axioms: 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767022.png" /> is a group; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767023.png" /> is a lattice; and 5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767025.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767026.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767027.png" /> is the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767028.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767030.png" />). In lattice-ordered groups, the following relations hold:
+
'' $  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/l057/l057670/l05767031.png" /></td> </tr></table>
+
A [[Group|group]]  $  G $
 +
on the set of elements of which a partial-order relation  $  \leq  $
 +
is defined possessing the properties: 1)  $  G $
 +
is a [[Lattice|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|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 $.
  
<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/l057/l057670/l05767032.png" /></td> </tr></table>
+
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 $(
 +
respectively,  $  x  ^ {+} = x\lor e $
 +
and  $  x  ^ {-} = x \wedge e $).
 +
In lattice-ordered groups, the following relations hold:
  
<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/l057/l057670/l05767033.png" /></td> </tr></table>
+
$$
 +
= x  ^ {+} x  ^ {-} ,\ \
 +
| x |  ^ {-} 1  \leq  x  \leq  | x | ,
 +
$$
  
Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767035.png" /> are called orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767036.png" />. Orthogonal elements commute.
+
$$
 +
| x |  = x  ^ {+} ( x  ^ {-} )  ^ {-} 1 ,\  x  ^ {+} \wedge ( x  ^ {-} )  ^ {-} 1  = e,
 +
$$
  
A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767037.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767038.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767039.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767041.png" />-subgroup if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767042.png" /> is a subgroup and a sublattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767043.png" />; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767044.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767045.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767047.png" />-ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767048.png" /> if it is normal and convex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767049.png" />. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767050.png" />-subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767051.png" />-ideals of a lattice-ordered group is distributive. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767053.png" />-homomorphism of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767054.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767055.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767056.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767057.png" /> is a [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767058.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767059.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767060.png" /> such that
+
$$
 +
( x \lor y)  ^ {-} 1  = x  ^ {-} 1 \wedge y  ^ {-} 1 ,\ \
 +
( x \wedge y)  ^ {-} 1  = x  ^ {-} 1 \lor y  ^ {-} 1 .
 +
$$
  
<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/l057/l057670/l05767061.png" /></td> </tr></table>
+
Two elements  $  x $
 +
and  $  y $
 +
are called orthogonal if  $  | x | \lor | y | = e $.  
 +
Orthogonal elements commute.
  
The kernels of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767062.png" />-homomorphisms are precisely the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767063.png" />-ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767064.png" />-groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767065.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767066.png" />-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767067.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767068.png" /> is a convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767069.png" />-subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767070.png" /> (cf. [[Convex subgroup|Convex subgroup]]).
+
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|homomorphism]] $  \phi $
 +
of the group  $  G $
 +
into the group  $  H $
 +
such that
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767071.png" /> of one-to-one order-preserving mappings of a totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767072.png" /> onto itself is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767073.png" />-group (if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767074.png" /> one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767075.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767076.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767077.png" />). Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767078.png" />-group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767079.png" />-isomorphic to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767080.png" />-subgroup of the lattice-ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767081.png" /> for a suitable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767082.png" />.
+
$$
 +
\phi ( x \lor y)  = \phi ( x) \lor \phi ( y) ,\ \
 +
\phi ( x \wedge y)  = \phi ( x) \wedge \phi ( y).
 +
$$
  
The class of all lattice-ordered groups is a variety of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767083.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057670/l05767085.png" />-groups, cf. also [[Totally ordered group|Totally ordered group]]).
+
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|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|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====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,  T. Feil,  "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,  T. Feil,  "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR></table>

Revision as of 22:15, 5 June 2020


$ 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=47589
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article