On Minimal Generating Sets of Certain Subsemigroups of Isometries

Year 2020, Volume: 8 Issue: 2, 71 - 78, 15.10.2020

Leyla Bugay Melek Yağcı

https://doi.org/10.36753/mathenot.723297

Abstract

Let $DP_{n}$ and $ODP_{n}$ be the semigroups of all isometries and of all order-preserving isometries on $X_{n}$,
respectively. In this paper we investigate the structure of minimal generating sets of the subsemigroup
$DP_{n,r}$= {α ∈ DPn : |im (α)| ≤ r} (similarly of the subsemigroup $ODP_{n,r}$ = {α ∈ ODPn : |im (α)| ≤ r})
for 2 ≤ r ≤ n − 1.                                                                                                                                                                               .

Keywords

Isometry, order-preserving /order-reversing map, (minimal) generating set

References

• [1] Al-Kharousi, F., Kehinde, R., Umar, A.: On the semigroup of partial isometries of a finite chain. Comm. Algebra. 44, 639–647 (2016).
• [2] Bugay, L., Yağcı M., Ayık, H.: The ranks of certain semigroups of partial isometries. Semigroup Forum. 97, 214–222 (2018).
• [3] Ganyushkin, O., Mazorchuk, V.: Classical finite transformation semigroups. Springer-Verlag. London (2009).
• [4] Howie, J. M.: Fundamentals of semigroup theory. Oxford University Press. New York (1995).
• [5] Wilson, R.J., Watkins, J.J.: Graphs, An Introductory Approach, A First Course in Discrete Mathematics. Jon Wiley & Sons Inc. Toronto (1990).