# Prime ideal theorem for double Boolean algebras

• Volume: 27, Issue: 2, page 263-275
• ISSN: 1509-9415

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## Abstract

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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 ${x}^{⊲}\in F$. 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).

## How to cite

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Léonard Kwuida. "Prime ideal theorem for double Boolean algebras." Discussiones Mathematicae - General Algebra and Applications 27.2 (2007): 263-275. <http://eudml.org/doc/276866>.

@article{LéonardKwuida2007,
abstract = {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).},
author = {Léonard Kwuida},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {double Boolean algebra; protoconcept algebra; concept algebra; weakly dicomplemented lattices; formal concept analysis},
language = {eng},
number = {2},
pages = {263-275},
title = {Prime ideal theorem for double Boolean algebras},
url = {http://eudml.org/doc/276866},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Léonard Kwuida
TI - Prime ideal theorem for double Boolean algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2007
VL - 27
IS - 2
SP - 263
EP - 275
AB - 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).
LA - eng
KW - double Boolean algebra; protoconcept algebra; concept algebra; weakly dicomplemented lattices; formal concept analysis
UR - http://eudml.org/doc/276866
ER -

## References

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1. [1] G. Boole, An investigation into the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities, Macmillan 1854, reprinted by Dover Publ. New York 1958.
2. [2] C. Herrmann, P. Luksch, M. Skorsky and R. Wille, Algebras of semiconcepts and double Boolean algebras, J. Heyn Klagenfurt, Contributions to General Algebra 13 (2001), 175-188. Zbl0986.03049
3. [3] B. Ganter and R. Wille, Formal Concept Analysis. Mathematical Foundations, Springer 1999. Zbl0909.06001
4. [4] L. Kwuida, Dicomplemented Lattices. A Contextual Generalization of Boolean Algebras, Shaker Verlag 2004. Zbl1183.06001
5. [5] R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (Ed.) Ordered Sets Reidel (1982), 445-470.
6. [6] R. Wille, Boolean Concept Logic, LNAI 1867 Springer (2000), 317-331. Zbl0973.03035
7. [7] R. Wille, Boolean Judgement Logic, LNAI 2120 Springer (2001), 115-128. Zbl0994.03025
8. [8] R. Wille, Preconcept algebras and generalized double Boolean algebras, LNAI 2961 Springer (2004), 1-13. Zbl1198.03084

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