The authors investigate how many informations on a semigroup \(S\) are given by the partial algebra \(E(S)\) of the idempotents of \(S\) where a product of \((e,f)\) is defined iff, under the multiplication of \(S\), \(ef\) or \(fe\) is equal to one of its factors. \(E(S)\) is the biordered set of \(S\). They introduce two representations \(\phi^ 0\) of \(S\) and \(\phi\) of \(E\) into the direct product \(T\times T^*\) of transformation semigroups \(T\) and \(T^*\). Here \(\phi^ 0s\to (\rho^ 0_ s,\lambda^ 0_ s)\) where \(T\) consists of the transformation semigroup of the set of Green's \({\mathcal L}\)-classes of \(S\) augmented by the symbol \(\infty\) and \(\rho^ 0_ s\) maps \(L^ 0_ x\) to \(L^ 0_{xs}\) if \(x{\mathcal R}_ 0xs\) is valid and to \(\infty\) otherwise. \(\lambda^ 0_ s\) is defined for \(T^*\), formed similarly for Green's \({\mathcal R}\)-classes. The other representation \(\phi\) maps \(e\to\phi_ e=(\rho_ e,\lambda_ e)\) for \(e\in E\). Here \(\hat L_ x\) consists of those \(y\in E\) which satisfy \(xy=x\) and \(yx=y\) and it is mapped under \(\rho_ e\) to \(\hat L_{xe}\) if \(ex'=x'\) for an \(x'\in\hat L_ x\) and to \(\infty\) otherwise. Similarly \(\lambda_ e\) is defined. With the notation \(\langle E\rangle\) for the subsemigroup of \(S\) generated by the set \(E\) the main theorem reads: \(\langle E\phi\rangle\) is isomorphic to \(\langle E\rangle\phi^ 0=\langle E\phi^ 0\rangle\) under \(\phi_{e_ 1}\phi_{e_ 2}\cdots\phi_{e_ n}\to\phi^ 0_{e_ 1e_ 2\cdots e_ n}\). Observe the case \(\langle E\rangle=S\), i.e. if \(S\) is idempotent generated. The concept of biordered sets has been introduced by \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)] and similarly, by \textit{R. E. Hartwig} [Math. Jap. 25, 1-13 (1980; Zbl 0442.06006)]. The present authors show: Any biordered set \(E\) is the biordered set of all idempotents of some semigroup \(S\), with the property that any \((\ker\phi^ 0)\)-class contains an idempotent in \(S\), if and only if \(E\phi =E(\langle E\phi\rangle)\) is valid.