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''lattice-ordered algebra''
 
''lattice-ordered algebra''
  
An [[Algebraic system|algebraic system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100102.png" /> over a totally ordered field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100104.png" /> is an associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100105.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100106.png" /> is a [[Lattice|lattice]] respect with the [[Partial order|partial order]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100107.png" /> and the following axioms hold:
+
An [[Algebraic system|algebraic system]] $  \{ A; \mathbf P , +, \cdot, \cle \} $
 +
over a totally ordered field $  \mathbf P $
 +
such that $  \{ A; \mathbf P , +, \cdot \} $
 +
is an associative algebra over $  \mathbf P $(
 +
cf. [[Associative rings and algebras|Associative rings and algebras]]), $  \{ A; \cle \} $
 +
is a [[Lattice|lattice]] respect with the [[Partial order|partial order]] $  \cle $
 +
and the following axioms hold:
  
1) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l1100108.png" />,
+
1) for all $  a,b,c \in A $,
  
<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/l110010/l1100109.png" /></td> </tr></table>
+
$$
 +
a \cle b \Rightarrow a + c \cle b + c,
 +
$$
  
2) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001010.png" />,
+
2) for all $  a,b,c \in A $,
  
<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/l110010/l11001011.png" /></td> </tr></table>
+
$$
 +
( c > 0 ) \& ( a \cle b ) \Rightarrow ( ac \cle bc ) \& ( ca \cle cb ) ,
 +
$$
  
3) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001013.png" />,
+
3) for all $  a,b \in A $
 +
and $  \alpha \in \mathbf P $,
  
<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/l110010/l11001014.png" /></td> </tr></table>
+
$$
 +
( \alpha > 0 ) \& ( a \cle b ) \Rightarrow ( \alpha a \cle \alpha c ) .
 +
$$
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001015.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001016.png" /> is called a strict <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001018.png" />-algebra if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001020.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001022.png" />. It is useful to describe an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001023.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001024.png" /> as an algebraic system of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001026.png" /> are the join and meet operations in the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001027.png" />.
+
An l $-
 +
algebra $  A $
 +
is called a strict l $-
 +
algebra if for $  a < b $
 +
and  $  c > 0 $
 +
one has $  ac < bc $,  
 +
$  ca < cb $.  
 +
It is useful to describe an l $-
 +
algebra $  \{ A; \mathbf P , +, \cdot, \cle \} $
 +
as an algebraic system of signature $  \{ \mathbf P, +, \cdot, \lor, \wedge \} $,  
 +
where $  \lor, \wedge $
 +
are the join and meet operations in the lattice $  \{ A, \cle \} $.
  
The most important examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001028.png" />-algebras are: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001029.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001030.png" /> of all continuous real-valued functions on a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001031.png" /> with respect to the natural operations and equipped with the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001032.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001033.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001035.png" />; and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001036.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001037.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001038.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001039.png" /> with order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001040.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001042.png" />.
+
The most important examples of l $-
 +
algebras are: the l $-
 +
algebra $  {\mathcal C} ( X, \mathbf R ) $
 +
of all continuous real-valued functions on a [[Topological space|topological space]] $  X $
 +
with respect to the natural operations and equipped with the order $  f \cle g $,  
 +
for $  f,g \in {\mathcal C} ( X, \mathbf R ) $,  
 +
if and only if $  f ( x ) \cle g ( x ) $
 +
for all $  x \in X $;  
 +
and the l $-
 +
algebra $  {\mathcal M} _ {n} ( \mathbf R ) $
 +
of all $  ( n \times n ) $-
 +
matrices over $  \mathbf R $
 +
with order $  \| {a _ {ij }  } \| \cle \| {b _ {ij }  } \| $
 +
if and only if $  a _ {ij }  \cle b _ {ij }  $
 +
for all $  i,j $.
  
A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001044.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001046.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001048.png" />-homomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001049.png" /> is a [[Homomorphism|homomorphism]] of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001051.png" /> and a homomorphism of the lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001053.png" />. The kernel of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001054.png" />-homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001055.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001057.png" />-ideal, i.e., an [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001058.png" /> that is also a convex sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001059.png" />.
+
A homomorphism $  \varphi : A \rightarrow B $
 +
of l $-
 +
algebras $  A $
 +
and $  B $
 +
is an l $-
 +
homomorphism if $  \varphi $
 +
is a [[Homomorphism|homomorphism]] of the algebras $  A $
 +
and $  B $
 +
and a homomorphism of the lattices $  A $
 +
and $  B $.  
 +
The kernel of an l $-
 +
homomorphism of $  A $
 +
is an l $-
 +
ideal, i.e., an [[Ideal|ideal]] of $  A $
 +
that is also a convex sublattice of $  A $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001061.png" /> is called the positive cone of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001063.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001064.png" />. For the positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001065.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001066.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001067.png" /> the following properties hold:
+
If $  P = \{ {x \in A } : {x \cge 0 } \} $,  
 +
then $  P $
 +
is called the positive cone of the l $-
 +
algebra $  A $.  
 +
For the positive cone $  P $
 +
of an l $-
 +
algebra $  A $
 +
the following properties hold:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001068.png" />;
+
1) $  P + P \subseteq P $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001069.png" />;
+
2) $  P \cap P = \{ 0 \} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001070.png" />;
+
3) $  P \cdot P \subseteq P $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001071.png" />;
+
4) $  \mathbf P  ^ {+} \cdot P \subseteq P $;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001072.png" /> is a lattice respect with the induced order. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001073.png" />. If, in an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001074.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001075.png" />, one can find a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001076.png" /> with the properties 1)–5), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001077.png" /> can be given the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001078.png" />-algebra with positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001079.png" /> by setting: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001080.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001081.png" />. It is correct to identify the order of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001082.png" />-algebra with its positive cone, and so an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001083.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001084.png" /> is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001085.png" />.
+
5) $  P $
 +
is a lattice respect with the induced order. Here, $  \mathbf P  ^ {+} = \{ {\alpha \in \mathbf P } : {\alpha \geq  0 } \} $.  
 +
If, in an algebra $  A $
 +
over $  \mathbf P $,  
 +
one can find a subset $  P $
 +
with the properties 1)–5), then $  A $
 +
can be given the structure of an l $-
 +
algebra with positive cone $  P $
 +
by setting: $  x \cle y \Rightarrow y - x \in P $
 +
for $  x,y \in A $.  
 +
It is correct to identify the order of an l $-
 +
algebra with its positive cone, and so an l $-
 +
algebra $  A $
 +
is often denoted by $  ( A,P ) $.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001086.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001087.png" /> is strict if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001089.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001090.png" />.
+
An l $-
 +
algebra $  ( A,P ) $
 +
is strict if and only if $  xy \neq 0 $
 +
for all $  x,y \in P $.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001091.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001092.png" /> is totally-ordered (an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001095.png" />-algebra) if its order is total (cf. also [[Totally ordered set|Totally ordered set]]). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001096.png" />-algebra is called an [[F-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001097.png" />-algebra]] if it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001098.png" />-subalgebra of the Cartesian product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l11001099.png" />-algebras. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010100.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010101.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010102.png" />-algebra if and only if there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010103.png" /> of total orders on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010105.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010106.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010107.png" />-algebras have been well investigated. Every Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010109.png" />-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010110.png" /> is commutative. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010111.png" /> of nilpotent elements in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010112.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010113.png" /> is a convex ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010114.png" /> and the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010115.png" /> has no zero divisors. There exists a full description of the finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010116.png" />-algebras. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010117.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010118.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010119.png" />-algebra if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010120.png" />,
+
An l $-
 +
algebra $  A $
 +
is totally-ordered (an $  o $-
 +
algebra) if its order is total (cf. also [[Totally ordered set|Totally ordered set]]). An l $-
 +
algebra is called an [[F-algebra| $  f $-
 +
algebra]] if it is an l $-
 +
subalgebra of the Cartesian product of 0 $-
 +
algebras. An l $-
 +
algebra $  ( A,P ) $
 +
is an $  f $-
 +
algebra if and only if there exists a set $  \{ {P _ {i} } : {i \in I } \} $
 +
of total orders on $  A $
 +
such that $  P = \cap _ {i \in I }  P _ {i} $.  
 +
$  o $-  
 +
and $  f $-
 +
algebras have been well investigated. Every Archimedean $  f $-
 +
algebra over $  \mathbf R $
 +
is commutative. The set $  N $
 +
of nilpotent elements in an $  o $-
 +
algebra $  A $
 +
is a convex ideal of $  A $
 +
and the quotient algebra $  A/N $
 +
has no zero divisors. There exists a full description of the finite-dimensional $  o $-
 +
algebras. An l $-
 +
algebra $  A $
 +
is an $  f $-
 +
algebra if and only if for all $  a,b,c \in A $,
  
<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/l110010/l110010121.png" /></td> </tr></table>
+
$$
 +
( a \wedge b = 0 ) \& ( c \cge 0 ) \Rightarrow ( ca \wedge b = 0 ) \& ( ac \wedge b = 0 ) .
 +
$$
  
The structure of the convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010122.png" />-subalgebras and prime ideals has been investigated.
+
The structure of the convex l $-
 +
subalgebras and prime ideals has been investigated.
  
The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110010/l110010123.png" />-algebras is used in the study of order-preserving linear transformations and orthomorphisms of ordered vector spaces (i.e., linear transformations preserving the orthogonality properties).
+
The theory of l $-
 +
algebras is used in the study of order-preserving linear transformations and orthomorphisms of ordered vector spaces (i.e., linear transformations preserving the orthogonality properties).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bigard,  K. Keimel,  S. Wolfenstein,  "Groupes et anneaux rétiqulés" , Springer  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bigard,  K. Keimel,  S. Wolfenstein,  "Groupes et anneaux rétiqulés" , Springer  (1977)</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


lattice-ordered algebra

An algebraic system $ \{ A; \mathbf P , +, \cdot, \cle \} $ over a totally ordered field $ \mathbf P $ such that $ \{ A; \mathbf P , +, \cdot \} $ is an associative algebra over $ \mathbf P $( cf. Associative rings and algebras), $ \{ A; \cle \} $ is a lattice respect with the partial order $ \cle $ and the following axioms hold:

1) for all $ a,b,c \in A $,

$$ a \cle b \Rightarrow a + c \cle b + c, $$

2) for all $ a,b,c \in A $,

$$ ( c > 0 ) \& ( a \cle b ) \Rightarrow ( ac \cle bc ) \& ( ca \cle cb ) , $$

3) for all $ a,b \in A $ and $ \alpha \in \mathbf P $,

$$ ( \alpha > 0 ) \& ( a \cle b ) \Rightarrow ( \alpha a \cle \alpha c ) . $$

An $ l $- algebra $ A $ is called a strict $ l $- algebra if for $ a < b $ and $ c > 0 $ one has $ ac < bc $, $ ca < cb $. It is useful to describe an $ l $- algebra $ \{ A; \mathbf P , +, \cdot, \cle \} $ as an algebraic system of signature $ \{ \mathbf P, +, \cdot, \lor, \wedge \} $, where $ \lor, \wedge $ are the join and meet operations in the lattice $ \{ A, \cle \} $.

The most important examples of $ l $- algebras are: the $ l $- algebra $ {\mathcal C} ( X, \mathbf R ) $ of all continuous real-valued functions on a topological space $ X $ with respect to the natural operations and equipped with the order $ f \cle g $, for $ f,g \in {\mathcal C} ( X, \mathbf R ) $, if and only if $ f ( x ) \cle g ( x ) $ for all $ x \in X $; and the $ l $- algebra $ {\mathcal M} _ {n} ( \mathbf R ) $ of all $ ( n \times n ) $- matrices over $ \mathbf R $ with order $ \| {a _ {ij } } \| \cle \| {b _ {ij } } \| $ if and only if $ a _ {ij } \cle b _ {ij } $ for all $ i,j $.

A homomorphism $ \varphi : A \rightarrow B $ of $ l $- algebras $ A $ and $ B $ is an $ l $- homomorphism if $ \varphi $ is a homomorphism of the algebras $ A $ and $ B $ and a homomorphism of the lattices $ A $ and $ B $. The kernel of an $ l $- homomorphism of $ A $ is an $ l $- ideal, i.e., an ideal of $ A $ that is also a convex sublattice of $ A $.

If $ P = \{ {x \in A } : {x \cge 0 } \} $, then $ P $ is called the positive cone of the $ l $- algebra $ A $. For the positive cone $ P $ of an $ l $- algebra $ A $ the following properties hold:

1) $ P + P \subseteq P $;

2) $ P \cap P = \{ 0 \} $;

3) $ P \cdot P \subseteq P $;

4) $ \mathbf P ^ {+} \cdot P \subseteq P $;

5) $ P $ is a lattice respect with the induced order. Here, $ \mathbf P ^ {+} = \{ {\alpha \in \mathbf P } : {\alpha \geq 0 } \} $. If, in an algebra $ A $ over $ \mathbf P $, one can find a subset $ P $ with the properties 1)–5), then $ A $ can be given the structure of an $ l $- algebra with positive cone $ P $ by setting: $ x \cle y \Rightarrow y - x \in P $ for $ x,y \in A $. It is correct to identify the order of an $ l $- algebra with its positive cone, and so an $ l $- algebra $ A $ is often denoted by $ ( A,P ) $.

An $ l $- algebra $ ( A,P ) $ is strict if and only if $ xy \neq 0 $ for all $ x,y \in P $.

An $ l $- algebra $ A $ is totally-ordered (an $ o $- algebra) if its order is total (cf. also Totally ordered set). An $ l $- algebra is called an $ f $- algebra if it is an $ l $- subalgebra of the Cartesian product of $ 0 $- algebras. An $ l $- algebra $ ( A,P ) $ is an $ f $- algebra if and only if there exists a set $ \{ {P _ {i} } : {i \in I } \} $ of total orders on $ A $ such that $ P = \cap _ {i \in I } P _ {i} $. $ o $- and $ f $- algebras have been well investigated. Every Archimedean $ f $- algebra over $ \mathbf R $ is commutative. The set $ N $ of nilpotent elements in an $ o $- algebra $ A $ is a convex ideal of $ A $ and the quotient algebra $ A/N $ has no zero divisors. There exists a full description of the finite-dimensional $ o $- algebras. An $ l $- algebra $ A $ is an $ f $- algebra if and only if for all $ a,b,c \in A $,

$$ ( a \wedge b = 0 ) \& ( c \cge 0 ) \Rightarrow ( ca \wedge b = 0 ) \& ( ac \wedge b = 0 ) . $$

The structure of the convex $ l $- subalgebras and prime ideals has been investigated.

The theory of $ l $- algebras is used in the study of order-preserving linear transformations and orthomorphisms of ordered vector spaces (i.e., linear transformations preserving the orthogonality properties).

References

[a1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[a2] A. Bigard, K. Keimel, S. Wolfenstein, "Groupes et anneaux rétiqulés" , Springer (1977)
How to Cite This Entry:
L-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-algebra&oldid=12581
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article