trace {relations}

Traces of Endorelations

Description

Compute the left or right trace of an endorelation.

Usage

relation_trace(x, which)

Arguments

x

an endorelation.

which

one of "left" or "right", or a unique abbreviation thereof.

Details

Let R be a crisp endorelation. The left and right trace of R contain all pairs x, y for which z R x implies z R y for all z (left trace) or y R z implies x R z for all z (right trace), respectively. These are the largest (in the natural ordering of relations with the same domain) relations such that R * S \le R or S * R \le R, respectively (where * denotes composition). In the fuzzy case, the memberships of the traces can be defined as the infima over the corresponding fuzzy membership implications. See Chapter 2.3 in Fodor and Roubens (1994) for more information.

References

J. Fodor and M. Roubens (1994), Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers: Dordrecht.