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

2022/08/05 by Apatsara Sareeto, Sareeto, Apatsara, Jörg Koppitz +1
Computer Science · #20M05 #20M18 #20M20 #F.4 #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2208.03202

openalex publication_date 2022/08/05 · openalex created_date 2022/08/09 · openalex updated_date 2026/07/28

Abstract

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. In this article, we step forward the study of a submonoid of the symmetric inverse semigroup. We explore the monoid of all order-, fence-, and parity-preserving transformations on an n-element chain. We also characterize the transformations in that monoid and show that it has a rank 3n-6. In particular, we provide a generating set A_n of minimal size and exhibit concrete normal forms for the transformations generated by A_n.

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