Commutative nilpotent transformation semigroups (Q6591554)

For a set \(X\), denote by \(\mathcal{T}(X)\) the semigroup of full transformations over \(X\). A semigroup \(S\) with a zero is said to be \textit{nilpotent} if there exists \(m\in\mathbb{N}\) such that the product of any \(m\) elements of \(S\) is equal to the zero. When \(m\) can be taken to be \(2\), the semigroup is said to be a \textit{null semigroup}.\N\NThe authors determine the maximum size of a commutative nilpotent subsemigroup of \(\mathcal{T}(X)\) when \(X\) is finite and specify which semigroups achieve this maximum size. These results are similar to the ones by \textit{P. J. Cameron} et al. [Comb. Theory 3, No. 3, Paper No. 16, 48 p. (2023; Zbl 1535.20297)] who obtained the maximum size of a null subsemigroup of \(\mathcal{T}(X)\) and characterized the null semigroups of maximum size.\N\NThe main result in the paper is Theorem 3.7, which states that for any given commutative nilpotent subsemigroup of \(\mathcal{T}(X)\) whose zero has rank \(1\), there exists a null semigroup with the same size. Allied to the referred results by \textit{P. J. Cameron} et al. [loc.cit.] one gets to know the maximum size of a commutative nilpotent subsemigroup of \(\mathcal{T}(X)\).\N\NBesides the importance on its own, Theorem 3.7 is crucial in proving other results, namely Theorem 3.11, which examines the size of the semigroups when the zero has rank at least \(2\) and Theorem 3.12, showing that a commutative nilpotent subsemigroup of \(\mathcal{T}(X)\) of maximum size is a null semigroup.\N\NThe proof of Theorem 3.7 is long and involving. The authors provide a quite helpful example that helps the reader follow it, which largely contributes, from my point of view, to making the reading of this paper very pleasant.