On the semigroup of all partial fence-preserving injections on a finite set

Abstract

For [Formula: see text], let [Formula: see text] be an [Formula: see text]-element set and let [Formula: see text] be a fence, also called a zigzag poset. As usual, we denote by [Formula: see text] the symmetric inverse semigroup on [Formula: see text]. We say that a transformation [Formula: see text] is fence-preserving if [Formula: see text] implies that [Formula: see text], for all [Formula: see text] in the domain of [Formula: see text]. In this paper, we study the semigroup [Formula: see text] of all partial fence-preserving injections of [Formula: see text] and its subsemigroup [Formula: see text]. Clearly, [Formula: see text] is an inverse semigroup and contains all regular elements of [Formula: see text] We characterize the Green’s relations for the semigroup [Formula: see text]. Further, we prove that the semigroup [Formula: see text] is generated by its elements with rank [Formula: see text]. Moreover, for [Formula: see text], we find the least generating set and calculate the rank of [Formula: see text].

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

“…We suppose now that n ≥ 3. I. Dimitrova and J. Koppitz proved F I m = J m for all natural numbers m in [4], which comprises several pages and a few of lemmas in [4]. For the case that n is odd, one can shorten the proof.…”

“…PAut(n, ) = F I n . This inverse semigroup was first studied by I. Dimitrova and J. Koppitz in [4]. The authors described the Green's relations on F I n .…”

“…rankα = |imα|. For the case that n is even, it is proved that rankF I n = n + 1 and a concrete generating set of F I n with n + 1 elements is given in [4]. On the other hand, the rank of F I n is still an open problem, whenever n is odd.…”

The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence

“…So-called coregular elements of this monoids were determined in [15]. On the other hand, in [6] Dimitrova and Koppitz investigated the monoid of all partial permutations preserving a zig-zag order on a set with n elements, by studing Green's relations and generating sets of this monoid.…”

GENERATING SETS OF SEMIGROUPS OF PARTIAL TRANSFORMATIONS PRESERVING A ZIG-ZAG ORDER ON $\mathbb{N}$

“…Moreover, Dimitrova and Koppitz have investigated the monoid of all order-preserving partial injections on an n-element fence. They determined the rank, whenever n is even [2]. Note that the monoid of all order-preserving (full) transformations on a fence is not regular.…”