# 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)
How to Cite This Entry:
L-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-algebra&oldid=47544
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article