UNSPECIFIED (1998) Consistency of natural relations on sets. COMBINATORICS PROBABILITY & COMPUTING, 7 (3). pp. 281-293. ISSN 0963-5483

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Abstract

The natural relations for sets are those definable in terms of the emptiness of the subsets corresponding to Boolean combinations of the sets. For pairs of sets, there are just five natural relations of interest, namely, strict inclusion in each direction, disjointness, intersection with the universe being covered, or not. Let N denote {1,2,...,n} and ((N)(2)) denote {(i,j) \ i, j is an element of N and i < j}. A function mu on ((N)(2)) specifies one of these relations for each pair of indices. Then mu is said to be consistent on M subset of or equal to N if and only if there exists a collection of sets corresponding to indices in M such that the relations specified by mu hold between each associated pair of the sets. Firstly, it is proved that if mu is consistent on all subsets of N of size three then mu is consistent on N. Secondly, explicit conditions that make mu consistent on a subset of size three are given as generalized transitivity laws. Finally, it is shown that the result concerning binary natural relations can be generalized to r-ary natural relations for arbitrary r greater than or equal to 2.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software Q Science > QA Mathematics
Journal or Publication Title:	COMBINATORICS PROBABILITY & COMPUTING
Publisher:	CAMBRIDGE UNIV PRESS
ISSN:	0963-5483
Date:	September 1998
Volume:	7
Number:	3
Number of Pages:	13
Page Range:	pp. 281-293
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/14960

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