On Semigroups of Orientation-preserving Transformations with Restricted Range

Abstract

Let X _n be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫 _n be the monoid of all orientation-preserving transformations of X _n . In this article, for any nonempty subset Y of X _n , we consider the subsemigroup 𝒪𝒫 _n (Y) of 𝒪𝒫 _n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫 _n (Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫 _n (Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛ _n of all either orientation-preserving or orientation-reversing transformations of X _n .

Key Words:

• Orientation-preserving
• Orientation-reversing
• Rank
• Restricted range
• Transformations

2010 Mathematics Subject Classification:

• 20M20
• 20M10

ACKNOWLEDGMENT

We wish to thank the anonymous referee for her/his valuable suggestions.

Notes

Communicated by V. Gould.