Almost f-algebras and d-algebras

Abstract

In this paper we study the classes of almost f-algebras and d-algebras. Apart from a survey of known properties we present some ‘new’ properties of almost f-algebras and d-algebras and we consider their connection with f-algebras. To be more precise, we show, among other things, in an elementary intrinsic manner, that every Archimedean almost f-algebra is commutative. This result is a considerable improvement upon the fact that every Archimedean f-algebra is commutative. Furthermore, we give a description of the set of nilpotent elements in both an Archimedean d-algebra and an Archimedean almost f-algebra.

The Range Inclusion Results for Algebraic Nil Derivations on Commutative and Noncommutative Algebras

“…For more information about this field, one can refer to [1,3]. A (real) algebra A which is simultaneously a vector lattice such that the partial ordering and the multiplication in A are compatible, that is a, b ∈ A + implies ab ∈ A + is called lattice-ordered algebra( briefly -algebra).…”

“…The -algebra A is said to be a d-algebra whenever a ∧ b = 0 in A implies ac ∧ bc = ca ∧ cb = 0, for all 0 ≤ c ∈ A. In general, d-algebras are not commutative, see [3].…”

On the Structure of Almost F-Algebras

“…On the other hand, it follows from Proposition 1 that a (A & ) p is a vector sublattice of B with a as positive cone. Therefore, there exists 0< he A 6 such that a (a) a (f3) -a (/) a (g) = a (h) 2 . Hence, by (4.2), we get and the theorem follows.…”

“…A d-algebra which is commutative or has positive squares is an almost /-algebra. More about almost /-algebras and (¿-algebras can be found in [2], [9] and we refer to [11] for basic concepts concerning /-algebras.…”

Extensions of orthosymmetric lattice bimorphisms

“…Almost f -algebras and d-algebras are not as well behaved and are less commonly found in applications. However, Archimedean almost f -algebras also are commutative (see [3], [4] and [8]), and their multiplication can be extended, though not uniquely to the Dedekind completion (see [4]). …”