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 \(\mathscr {FI}_{n}\) of all order-preserving partial injections on an n-element fence. In particular, we provide a minimal generating set for \(\mathscr {FI}_{n}\). In the present paper, n is odd since this problem for even n was already solved by I. Dimitrova and J. Koppitz.

中文翻译：

---

有限栅栏上部分自同构的逆半群的秩

围栏是（有限）集合上特定的局部顺序，接近线性顺序。在本文中，我们计算了n元栅栏上所有保留顺序的部分注入的半群\（\ mathscr {FI} _ {n} \）的秩。特别是，我们为\（\ mathscr {FI} _ {n} \）提供了最小生成集。在本文中，n是奇数，因为偶数n的问题已经由I. Dimitrova和J. Koppitz解决。