Research articles
ScienceAsia 50 (2024):ID 2024011 112 doi:
10.2306/scienceasia15131874.2024.011
Green’s relations and natural partial order on BaerLevi
semigroups of partial transformations with restricted range
ABSTRACT: Let X be an infinite set and I(X) the symmetric inverse semigroup on X. For a nonempty subset Y of X
and an infinite cardinal q such that X ? q, let PS(X, Y, q) = {? ? I(X) : XX? = q and X? ? Y }. Then PS(X, Y, q) is
a generalization of the partial BaerLevi semigroup PS(X, q) = {? ? I(X) : XX? = q} which has been studying since
1975. In this paper, we describe the Green?s relations and characterize the natural partial order on PS(X, Y, q). With
respect to this partial order, we determine when two elements are related, find all the maximum, minimum, maximal,
minimal, lower cover and upper cover elements. Also, we describe elements which are compatible and we investigate
the greatest lower bound and the least upper bound of two elements in PS(X, Y, q).
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^{a} 
Department of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University,
Chiang Mai 50180 Thailand 
* Corresponding author, Email: boorapa_sin@cmru.ac.th
Received 13 May 2022, Accepted 28 Oct 2023
