Geodesic
Prime ideal theorem for double Boolean algebras
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 263-275

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Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or x^⊲ ∈ F. In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G ∩ I = ∅ (i.e. the Prime Ideal Theorem for double Boolean algebras).
Keywords: double Boolean algebra, protoconcept algebra, concept algebra, weakly dicomplemented lattices 
Kwuida, Léonard. Prime ideal theorem for double Boolean algebras. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 263-275. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a6/
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