L-algebra

lattice-ordered algebra

An algebraic system over a totally ordered field such that is an associative algebra over (cf. Associative rings and algebras), is a lattice respect with the partial order and the following axioms hold:

1) for all ,

2) for all ,

3) for all and ,

An -algebra is called a strict -algebra if for and one has , . It is useful to describe an -algebra as an algebraic system of signature , where are the join and meet operations in the lattice .

The most important examples of -algebras are: the -algebra of all continuous real-valued functions on a topological space with respect to the natural operations and equipped with the order , for , if and only if for all ; and the -algebra of all -matrices over with order if and only if for all .

A homomorphism of -algebras and is an -homomorphism if is a homomorphism of the algebras and and a homomorphism of the lattices and . The kernel of an -homomorphism of is an -ideal, i.e., an ideal of that is also a convex sublattice of .

If , then is called the positive cone of the -algebra . For the positive cone of an -algebra the following properties hold:

1) ;

2) ;

3) ;

4) ;

5) is a lattice respect with the induced order. Here, . If, in an algebra over , one can find a subset with the properties 1)–5), then can be given the structure of an -algebra with positive cone by setting: for . It is correct to identify the order of an -algebra with its positive cone, and so an -algebra is often denoted by .

An -algebra is strict if and only if for all .

An -algebra is totally-ordered (an -algebra) if its order is total (cf. also Totally ordered set). An -algebra is called an -algebra if it is an -subalgebra of the Cartesian product of -algebras. An -algebra is an -algebra if and only if there exists a set of total orders on such that . - and -algebras have been well investigated. Every Archimedean -algebra over is commutative. The set of nilpotent elements in an -algebra is a convex ideal of and the quotient algebra has no zero divisors. There exists a full description of the finite-dimensional -algebras. An -algebra is an -algebra if and only if for all ,

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

The theory of -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)