Bulletin of the
Korean Mathematical Society BKMS

In this paper, we construct nilpotent semigroups $S$ such that $S^n=\{0\}$, $S^{n-1}\not=\{0\}$ and $\G(S)$ is a refinement of the star graph $K_{1,n-3}$ with center $c$ together with finitely many or infinitely many end vertices adjacent to $c$, for each finite positive integer $n\ge 5$. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpotent semigroups when $n=5,\, 6$ and in finite cases.

Keywords: nilpotent semigroup, refinement of a star graph, structure, counting formula

MSC numbers: Primary 20M14; Secondary 05C90