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

Ilinka Dimitrova

Faculty of Mathematics and Natural Science, South-West University “Neofit Rilski”, 2700 Blagoevgrad, Bulgaria

E-mail Address: ilinka_dimitrova@swu.bg

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and

Jörg Koppitz

Institute of Mathematics, University of Potsdam, 14476 Potsdam, Germany

E-mail Address: koppitz@uni-potsdam.de

Corresponding author.

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https://doi.org/10.1142/S0219498817502231Cited by:11 (Source: Crossref)

Abstract

For $n \in ℕ$ , let $X_{n} = {a_{1}, a_{2}, \dots, a_{n}}$ be an $n$ -element set and let $F = (X_{n}; <_{f})$ be a fence, also called a zigzag poset. As usual, we denote by $I_{n}$ the symmetric inverse semigroup on $X_{n}$ . We say that a transformation $α \in I_{n}$ is fence-preserving if $x <_{f} y$ implies that $x α <_{f} y α$ , for all $x, y$ in the domain of $α$ . In this paper, we study the semigroup $P F I_{n}$ of all partial fence-preserving injections of $X_{n}$ and its subsemigroup $I F_{n} = {α \in P F I_{n} : α^{- 1} \in P F I_{n}}$ . Clearly, $I F_{n}$ is an inverse semigroup and contains all regular elements of $P F I_{n} .$ We characterize the Green’s relations for the semigroup $I F_{n}$ . Further, we prove that the semigroup $I F_{n}$ is generated by its elements with rank $\geq n - 2$ . Moreover, for $n \in 2 ℕ$ , we find the least generating set and calculate the rank of $I F_{n}$ .

Communicated by L. Bokut

Keywords:

AMSC: 20M20, 20M18

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