The rank of the inverse semigroup of partial automorphisms on a finite fence

Abstract: A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $\mathcal{FI}{n}$ of all order-preserving partial injections on an $n$-element fence. In particular, we provide a minimal generating set for $\mathcal{FI}{n}$. In the present paper, $n$ is odd since this problem for even $n$ was already solved by I. Dimitrova and J. Koppitz.