Natural Partial Order on the Semigroups of Partial Isometries of a Finite Chain

Chaiwat Namnak, Ekkachai Laysirikul, Nares Sawatraksa

Abstract

Let In denote the 1 − 1 partial transformation semigroup on a set
{1, 2, . . . , n} and let DPn = {α ∈ In : ∀x, y ∈ Dom α, |xα − yα| = |x − y|} and
ODPn = {α ∈ DPn : ∀x, y ∈ Dom α, x ≤ y ⇒ xα ≤ yα}. Then DPn and ODPn
are subsemigroups of In. The purpose of this research, we study the natural partial
orders on DPn and ODPn and characterize when two elements of DPn and ODPn
are related under this partial order. Moreover, we give a necessary and sufficient
conditions for elements in DPn and ODPn to be maximal or minimal elements.