Prime ideal theorem for double Boolean algebras

• Autorzy: Léonard Kwuida
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

double Boolean algebra; protoconcept algebra; concept algebra; weakly dicomplemented lattices

Kategorie tematyczne

• 06E05: Structure theory

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2007
• Tom: 27
• Numer: 2
• Strony: 263-275

Daty

wydano

2007

otrzymano

2006-07-24

poprawiono

2006-12-13

Twórcy

autor

Léonard Kwuida

• Universität Bern Mathematisches Institut, Sidlerstr. 5, CH-3012 Bern

Bibliografia

• [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] 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.
• [3] B. Ganter and R. Wille, Formal Concept Analysis. Mathematical Foundations, Springer 1999.
• [4] L. Kwuida, Dicomplemented Lattices. A Contextual Generalization of Boolean Algebras, Shaker Verlag 2004.
• [5] R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (Ed.) Ordered Sets Reidel (1982), 445-470.
• [6] R. Wille, Boolean Concept Logic, LNAI 1867 Springer (2000), 317-331.
• [7] R. Wille, Boolean Judgement Logic, LNAI 2120 Springer (2001), 115-128.
• [8] R. Wille, Preconcept algebras and generalized double Boolean algebras, LNAI 2961 Springer (2004), 1-13.

Identyfikatory

DOI

10.7151/dmgaa.1130