| Summary: | Let $X$ be a set with infinite regular cardinality $m$ and let $\mathcal T(X)$ be the semigroup of all self-maps of $X$. The semigroup $Q_m$ of ‘balanced’ elements of $\mathcal T(X)$ plays an important role in the study by Howie [3,5,6] of idempotent-generated subsemigroups of $\mathcal T(X)$, as does the subset $S_m$ of ‘stable’ elements, which is a subsemigroup of $Q_m$ if and only if $m$ is a regular cardinal. The principal factor $P_m$ of $Q_m$, corresponding to the maximum $\mathcal J$-class $J_m$, contains $S_m$ and has been shown in [7] to have a number of interesting properties. Let $N_2$ be the set of all nilpotent elements of index 2 in $P_m$. Then the subsemigroup $<N_2>$ of $P_m$ generated by $N_2$ consists exactly of the elements in $P_m\backslash S_m$. Moreover $P_m\backslash S_m$ has 2-nilpotent-depth 3, in the sense that $N_2\cup N_2^2 \subset P_m\backslash S_m=N_2 \cup N_2^2\cup N_2^3$.
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