L-algebra

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)