# On finite elements in $$f$$ -algebras and in product algebras

On finite elements in $$f$$ -algebras and in product algebras Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $$\ell$$ -algebras, preferably in $$f$$ -algebras, and in product algebras. The additional structure of an associative multiplication leads to new questions and some new properties concerning the collections of finite, totally finite and self-majorizing elements. In some cases the order ideal of finite elements is a ring ideal as well. It turns out that a product of elements in an $$f$$ -algebra is a finite element if at least one factor is finite. If the multiplicative unit exists, the latter plays an important role in the investigation of finite elements. For the product of special $$f$$ -algebras an element is finite in the algebra if and only if its power is finite in the product algebra. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On finite elements in $$f$$ -algebras and in product algebras

22 pages

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### Abstract

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $$\ell$$ -algebras, preferably in $$f$$ -algebras, and in product algebras. The additional structure of an associative multiplication leads to new questions and some new properties concerning the collections of finite, totally finite and self-majorizing elements. In some cases the order ideal of finite elements is a ring ideal as well. It turns out that a product of elements in an $$f$$ -algebra is a finite element if at least one factor is finite. If the multiplicative unit exists, the latter plays an important role in the investigation of finite elements. For the product of special $$f$$ -algebras an element is finite in the algebra if and only if its power is finite in the product algebra.

### Journal

PositivitySpringer Journals

Published: Oct 13, 2012

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