Numerical representation of binary relations with a multiplicative error function (Q1857103)

The author considers binary relations which admit cardinal representation with error, i.e., \(P\leq A\times A\), \(u:A\to R\), \(\delta :A \times A\to R\) such that \(xPy\Leftrightarrow u(x)-u(y)>\delta(x,y)\). In the case \(\delta(x,y)= \alpha^2/u (x)u(y)\), it is shown that \(P\) is an interval order, if \(u\) is positive. Further, it is proved that \(P\) is a semiorder if and only if it admits a representation with \(\delta(x,y)=\alpha^2 u(x)u(y)\) and \(u\) is positive. We note that \(P\) is a semiorder iff \(\overline P\) is a semiorder, where \(\overline P\) is defined by \(x\overline P y\Leftrightarrow yPx\). Therefore, the previous result is equivalent to the following one: ``\(P\) is a semiorder if and only if it admits a numerical representation with a positive constant error''. This result has been obtained in the literature.