The rank of the semigroup of order-, fence-, and parity-preserving partial injections on a finite set

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let [Formula: see text] be a positive natural number and [Formula: see text] be the semigroup of all fence-preserving partial one-to-one maps of [Formula: see text] into itself with respect to composition of maps and the fence [Formula: see text]. There is considered the inverse semigroup [Formula: see text] of all [Formula: see text] such that [Formula: see text] is regular in [Formula: see text], order-preserving with respect to the order [Formula: see text] and parity-preserving. According to the main result of the paper, it is [Formula: see text] the least of the cardinalities of the generating sets of [Formula: see text] for [Formula: see text]. There is determined a concrete representation of a generating set of minimal size.