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Almost f-algebras and d-algebras. (English) Zbl 0707.06009

By definition, a lattice ordered algebra A is an f-algebra if \(a\bot b\) (that is, \(| a| \wedge | b| =0)\Rightarrow\) ca\(\bot b\) and ac\(\bot b\) for all a,b,c\(\in A\), an almost f-algebra if \(a\bot b\Rightarrow\) \(ab=0\) for all a,b\(\in A\), and a d-algebra if \(a\bot b\Rightarrow\) ac\(\bot bc\) for all a,b,c\(\in A\). Every f-algebra is an almost f-algebra and a d-algebra, but there are almost f-algebras that are not d-algebras and vice versa. The authors present algebraic proofs of several known properties of almost f-algebras. For example, it is shown that Archimedean almost f-algebras are commutative, and every element a in the nil-radical of an Archimedean f-algebra satisfies \(a^ 3=0\). The proofs in this paper are interesting even for f-algebras, because they do not use the representation theory.
Reviewer: R.S.Pierce

MSC:

06F25 Ordered rings, algebras, modules
46A40 Ordered topological linear spaces, vector lattices
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