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