Abstract
Double Boolean algebras were introduced in [Wi00a] as a variety fundamental for Boolean Concept Logic, an extension of Formal Concept Analysis allowing negations of formal concepts. In this paper, the free double Boolean algebra generated by the constants is described. Moreover, we show that every free double Boolean algebra with at least one generator is infinite. A measure of the complexity of terms specific for double Boolean algebras is introduced. This, together with a modification of the algorithm for protoconcept exploration (cf. [Vo04]) yields double Boolean algebras containing a counterexample to every term identity up to a given complexity if the identity does not hold in general. These algebras can be constructed automatically, thus the word problem for free double Boolean algebras is solved.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Burmeister, P.: ConImp – Ein Programm zur Fomalen Begriffsanalyse. In: Stumme, G., Wille, R. (eds.) Begriffliche Wissensverarbeitung: Methoden und Anwendungen, pp. 25–56. Springer, Heidelberg (2000)
Burris, S., Sankappanavar, H.P.: A course in universal algebra. Millenium Edition (2000), http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)
Herrmann, C., et al.: Algebras of Semiconcepts and Double Boolean Algebras. Contributions to General Algebra, vol. 13. Verlag Johannes Heyn, Klagenfurt (2000)
Skorsky, M.: Regular Monoids Generated by two Galois Connections. Semigroup Forum 39, 263–293 (1989)
Stumme, G.: Concept Exploration. Knowledge Acquisition in Conceptual Knowledge Systems. Shaker, Aachen (1997)
Vormbrock, B.: Kongruenzrelationen auf doppelt-booleschen Algebren. Diplomarbeit, FB Mathematik, TU Darmstadt (2002)
Vormbrock, B.: Congruence Relations on Double Boolean Algebras. Algebra Universalis (submitted)
Vormbrock, B.: A First Step Towards Protoconcept Exploration. In: Eklund, P.W. (ed.) ICFCA 2004. LNCS (LNAI), vol. 2961, pp. 208–221. Springer, Heidelberg (2004)
Vormbrock, B., Wille, R.: Semiconcept and Protoconcept Algebras: The Basic Theorems. In: Ganter, B., Stumme, G., Wille, R. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3626, Springer, Heidelberg (2005)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered sets, pp. 445–470. Reidel, Dordrecht (1982)
Wille, R.: Boolean Concept Logic. In: Ganter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS, vol. 1867, pp. 317–331. Springer, Heidelberg (2000)
Wille, R.: Contextual Logic Summary. In: Stumme, G. (ed.) Working with Conceptual Structures. Contributions to ICCS 2000, pp. 265–276. Shaker, Aachen (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Vormbrock, B. (2007). A Solution of the Word Problem for Free Double Boolean Algebras. In: Kuznetsov, S.O., Schmidt, S. (eds) Formal Concept Analysis. ICFCA 2007. Lecture Notes in Computer Science(), vol 4390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70901-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-70901-5_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70828-5
Online ISBN: 978-3-540-70901-5
eBook Packages: Computer ScienceComputer Science (R0)