Defining relations for non-permutational finite transformations (Q1279670)

Let \(I\) denote an arbitrary set and for \(i,j\in I\), denote by \([i/j]\) the transformation of \(I\) which maps \(i\) to \(j\) and fixes all other elements of \(I\). Such a transformation is referred to as a replacement on \(I\). Let \(T=\{[i/j]:i,j\in I\}\). The semigroup \(NP(I)\) which is generated by \(T\) is the semigroup of all nonpermutational finite transformations of I. \textit{R. J. Thompson} [Banach Cent. Publ. 28, 327-342 (1993; Zbl 0797.03062)] proved that \(NP(I)\) is presentable with finitely many schemas of defining relations. The author of the paper under review gives a simpler proof of this result. In addition, the length of the relations of his schemas is bounded and this settles a conjecture of \textit{I. Fleischer} [Algebra Univers. 33, No. 2, 186-190 (1995; Zbl 0821.03030)].