## 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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### References:

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