Prime Ideal Theorem for Double Boolean Algebras
نویسندگان
چکیده
Double Boolean algebras are algebras (D,u,t, , ,⊥,>) 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 u (resp. t). A filter F is called primary if F 6= ∅ 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).
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