Measuring Development by Asis Kumar Banerjee

Author:Asis Kumar Banerjee
Language: eng
Format: epub
ISBN: 9789811561610
Publisher: Springer Singapore

From an intuitive viewpoint, however, the cardinal approach to fuzzy relations is sometimes criticised for not being in conformity with the basic ideas of the fuzzy sets approach. If the basic point is that some statements are inherently ambiguous, then it is somewhat self-contradictory to specify the degree of ambiguity by giving to it a precise numerical value. For this reason, a somewhat different type of approach to fuzzy relations has been proposed. It is known as the theory of ordinally fuzzy relations. There are different versions of the theory. In the version that we adopted for our purposes in the text, we still assume, for convenience, that, for any x and y in X, R(x, y) is numerically specified: for any x and y in X, R(x, y) is still a real number. However, we do not perform arithmetic operations such as additions or multiplication on these numbers. For any x, y, z and w in X, all we care about is whether or not R(x, y) ≥ R(z, w). In other words, all we need to use is the natural order of real numbers given by the (crisp) relation ≥ on the real line.

R now is assumed to be a mapping from X2 into a bounded subset A of the real line with the usual order relation ≥ on the real line. Since A is bounded, it will have a supremum (a*, say) and an infimum (a*, say).

It may be noted in passing that the notion of an ordinal fuzzy relation formulated here is an example of what are called “L-fuzzy binary relations” in mathematics. An L-fuzzy binary relation S on a set B is a mapping from B × B into a lattice L. A lattice is any partially ordered set (not necessarily a set of real numbers) in which every pair of members has a least upper bound and a greatest lower bound with respect to the specified partial order relation (T, say). Salii (1965) (in Russian) contained an early exploration of the idea. Recent contributions in this area are developments based on Goguen (1967). It should be noted that in our framework an ordinal fuzzy relation is, trivially, a complete relation if completeness is defined to mean that, for any x and y in X, R(x, y) ≥ R(y, x) or R(y, x) ≥ R(x, y). An L-fuzzy relation S, however, would be complete if and only if S(x, y) T S(y, x) or S(y, x) T S(x, y). It would be a non-trivial restriction. An arbitrary L-fuzzy relation is not necessarily complete. In this book, we do not work in the more general L-fuzzy framework and confine ourselves to the notion of an ordinal fuzzy relation as formulated in the previous paragraph. For an application of the Goguen framework (in the context of a problem in social choice theory), see Barrett, Pattanaik and Salles (1992).

To continue with our formulation, an ordinal fuzzy relation R on X is called reflexive if, for any x in X, R(x, x) = a*. As noted above R is, by definition,

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