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 
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     title = {Prime ideal theorem for double {Boolean} algebras},
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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/