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''lattice-ordered algebra''

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  $  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|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|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  $  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|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 $,

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

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====
<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>