Compatible extensions of fuzzy relations

Abstract

In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999).

Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results.

In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan’s theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive.

Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter’s theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.

Keywords

Fuzzy relation / compatible extension / transitive-consistent

References

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