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About quotient and division of crisp and fuzzy relations.

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Title: About quotient and division of crisp and fuzzy relations.
Authors: Bosc, P.1 Patrick.Bosc@enssat.fr, Pivert, O.1 pivert@enssat.fr, Rocacher, D.1 rocacher@enssat.fr
Source: Journal of Intelligent Information Systems. Oct2007, Vol. 29 Issue 2, p185-210. 26p. 1 Diagram, 5 Charts.
Subjects: Relational databases, Fuzzy relational calculus, Fuzzy mathematics, Fuzzy sets, Fuzzy algorithms, Set theory, Database management, Boolean searching, Information retrieval
Abstract: The role and properties of the division are very well known in the context of queries addressed to regular relational databases. However, Boolean queries whose result is expressed in terms of all or nothing may turn out to be too limited to answer certain user needs and it is desirable to envisage extended queries by introducing preferences in the conditions. In this paper, two lines of extension of the division operator are studied: (i) operand relations of the division are fuzzy relations (i.e., they are made of weighted tuples) and (ii) the universal quantifier underlying the division is weakened. Various approaches to these extensions can be considered and one of our goals is to point out those which ensure that the resulting relation is a quotient (in reference to the characterization of the quotient of two integers). So doing, a sound semantics for the extended division is guaranteed. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Intelligent Information Systems is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Intelligent+Information+Systems%22">Journal of Intelligent Information Systems</searchLink>. Oct2007, Vol. 29 Issue 2, p185-210. 26p. 1 Diagram, 5 Charts.
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  Data: <searchLink fieldCode="DE" term="%22Relational+databases%22">Relational databases</searchLink><br /><searchLink fieldCode="DE" term="%22Fuzzy+relational+calculus%22">Fuzzy relational calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Fuzzy+mathematics%22">Fuzzy mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Fuzzy+sets%22">Fuzzy sets</searchLink><br /><searchLink fieldCode="DE" term="%22Fuzzy+algorithms%22">Fuzzy algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Set+theory%22">Set theory</searchLink><br /><searchLink fieldCode="DE" term="%22Database+management%22">Database management</searchLink><br /><searchLink fieldCode="DE" term="%22Boolean+searching%22">Boolean searching</searchLink><br /><searchLink fieldCode="DE" term="%22Information+retrieval%22">Information retrieval</searchLink>
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  Data: The role and properties of the division are very well known in the context of queries addressed to regular relational databases. However, Boolean queries whose result is expressed in terms of all or nothing may turn out to be too limited to answer certain user needs and it is desirable to envisage extended queries by introducing preferences in the conditions. In this paper, two lines of extension of the division operator are studied: (i) operand relations of the division are fuzzy relations (i.e., they are made of weighted tuples) and (ii) the universal quantifier underlying the division is weakened. Various approaches to these extensions can be considered and one of our goals is to point out those which ensure that the resulting relation is a quotient (in reference to the characterization of the quotient of two integers). So doing, a sound semantics for the extended division is guaranteed. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Intelligent Information Systems is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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