On the Nilpotent Rank of Partial Transformation Semigroups

G.U. Garba

Abstract: In [7] Sullivan proved that the semigroup $SP_{n}$ of all strictly partial transformations on the set $X_{n}=\{1,...,n\}$ is nilpotent-generated if $n$ is even, and that if $n$ is odd the nilpotents in $SP_{n}$ generate $SP_{n}\backslash W_{n-1}$ where $W_{n-1}$ consists of all elements in $[n-1,n-1]$ whose completions are odd permutations. We now show that whether $n$ is even or odd both the rank and the nilpotent rank of the subsemigroup of $SP_{n}$ generated by the nilpotents are equal to $n+2$.

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