On the isomorphism problem of concept algebras

Kwuida, Léonard; Machida, Hajime

doi:10.1007/s10472-010-9194-x

Please use this identifier to cite or link to this item: https://doi.org/10.21256/zhaw-29870

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	On the isomorphism problem of concept algebras
Authors:	Kwuida, Léonard Machida, Hajime
et. al:	No
DOI:	10.1007/s10472-010-9194-x 10.21256/zhaw-29870
Published in:	Annals of Mathematics and Artificial Intelligence
Volume(Issue):	59
Issue:	2
Page(s):	223
Pages to:	239
Issue Date:	2010
Publisher / Ed. Institution:	Springer
ISSN:	1012-2443 1573-7470
Language:	English
Subjects:	Concept algebras; Negation; Weakly dicomplemented lattices; Representation problem; Boolean algebras; Field of sets; Formal concept analysis
Subject (DDC):	510: Mathematics
Abstract:	Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on concepts. They have been introduced to capture the equational theory of concept algebras (Wille 2000; Kwuida 2004). They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in Kwuida (2004), is whether complete weakly dicomplemented lattices are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem 4). We also provide a new proof of a well known result due to M.H. Stone (Trans Am Math Soc 40:37–111, 1936), saying that each Boolean algebra is a field of sets (Corollary 4). Before these, we prove that the boundedness condition on the initial definition of weakly dicomplemented lattices (Definition 1) is superfluous (Theorem 1, see also Kwuida (2009)).
Further description:	Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)
URI:	https://digitalcollection.zhaw.ch/handle/11475/29870
Fulltext version:	Published version
License (according to publishing contract):	Licence according to publishing contract
Departement:	School of Engineering
Organisational Unit:	Institute of Applied Mathematics and Physics (IAMP)
Appears in collections:	Publikationen School of Engineering

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Kwuida, L., & Machida, H. (2010). On the isomorphism problem of concept algebras. Annals of Mathematics and Artificial Intelligence, 59(2), 223–239. https://doi.org/10.1007/s10472-010-9194-x

Kwuida, L. and Machida, H. (2010) ‘On the isomorphism problem of concept algebras’, Annals of Mathematics and Artificial Intelligence, 59(2), pp. 223–239. Available at: https://doi.org/10.1007/s10472-010-9194-x.

L. Kwuida and H. Machida, “On the isomorphism problem of concept algebras,” Annals of Mathematics and Artificial Intelligence, vol. 59, no. 2, pp. 223–239, 2010, doi: 10.1007/s10472-010-9194-x.

KWUIDA, Léonard und Hajime MACHIDA, 2010. On the isomorphism problem of concept algebras. Annals of Mathematics and Artificial Intelligence. 2010. Bd. 59, Nr. 2, S. 223–239. DOI 10.1007/s10472-010-9194-x

Kwuida, Léonard, and Hajime Machida. 2010. “On the Isomorphism Problem of Concept Algebras.” Annals of Mathematics and Artificial Intelligence 59 (2): 223–39. https://doi.org/10.1007/s10472-010-9194-x.

Kwuida, Léonard, and Hajime Machida. “On the Isomorphism Problem of Concept Algebras.” Annals of Mathematics and Artificial Intelligence, vol. 59, no. 2, 2010, pp. 223–39, https://doi.org/10.1007/s10472-010-9194-x.