On the semigroup of all partial fence-preserving injections on a finite set
- For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α−1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green’s relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n−2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.
| Author details: | Ilinka DimitrovaORCiDGND, Jörg KoppitzORCiDGND |
|---|---|
| DOI: | https://doi.org/10.1142/S0219498817502231 |
| ISSN: | 0219-4988 |
| ISSN: | 1793-6829 |
| Title of parent work (English): | Journal of Algebra and Its Applications |
| Publisher: | World Scientific |
| Place of publishing: | Singapore |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2017/01/12 |
| Publication year: | 2017 |
| Release date: | 2021/02/08 |
| Tag: | Finite transformation semigroup; Green´s Relations; fence-preserving transformations; generators; inverse semigroup; rank |
| Volume: | 16 |
| Issue: | 12 |
| Number of pages: | 14 |
| Funding institution: | Alexander von Humboldt Foundation |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
