Bifunctional-elementary relation algebras

Autores: Mohamed El Bachraoui
Localización: Algebra universalis, ISSN 0002-5240, Vol. 60, Nº. 4, 2009, págs. 425-438
Idioma: inglés
DOI: 10.1007/s00012-009-2121-5
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Resumen
- A relation algebra is bifunctional-elementary if it is atomic and for any atom a, the element a;1;a is the join of at most two atoms, and one of these atoms is bifunctional (an element x is bifunctional if $$x;x^{\smile} + x^{\smile} ;x \leq 1$$�). We show that bifunctional-elementary relation algebras are representable. Our proof combines the representation theorems for: pair-dense relation algebras given by R. Maddux; relation algebras generated by equivalence elements provided corresponding relativizations are representable by S. Givant; and strong-elementary relation algebras dealt with in our earlier work. It turns out that atomic pair-dense relation algebras are bifunctional elementary, showing that our theorem generalizes the representation theorem of atomic pair-dense relation algebras. The problem is still open whether the related classes of rather elementary, functional-elementary, and strong functional-elementary relation algebras are representable.