The inverse semigroup of all fence-preserving injections and its maximal subsemigroups

Abstract

In this paper, we study the inverse semigroup \(IF_{n}\) of all partial injections \(\alpha\) on an \(n\)-element set such that both \(\alpha\) and \(\alpha^{-1}\) are fence-preserving (preserve the zig-zag order). The main result of this paper is the characterization of the maximal subsemigroups of \(IF_{n}\): There are five types of maximal subsemigroups, whenever \(n\) is odd; if \(n\) is even, then the maximal semigroups are of the form \(IF_{n}\setminus \{\alpha \}\), where \(\alpha\) belongs to the least generating set of \(IF_{n}\). Moreover, we describe the i-conjugate elements in \(IF_{n}\).