Greenn’s relations and natural partial order on Baer-Levi 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 Baer-Levi 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, E-mail: boorapa_sin@cmru.ac.th

Received 13 May 2022, Accepted 28 Oct 2023