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relations (version 0.6-16)

trace: 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+Roubens:1994 for more information.

References

Fodor+Roubens:1994