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
@article{DMGAA_2007_27_2_a6,
author = {Kwuida, L\'eonard},
title = {Prime ideal theorem for double {Boolean} algebras},
journal = {Discussiones Mathematicae. General Algebra and Applications},
pages = {263--275},
publisher = {mathdoc},
volume = {27},
number = {2},
year = {2007},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a6/}
}
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/