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has been cited by the following article:

• TITLE: On the Number of Idempotent Partial Contraction Mappings of a Finite Chain

  AUTHORS: Oladapo Adekunle Ojo, Fatma Salim Ali Al-Kharousi, Abdullahi Umar

  KEYWORDS: Height, Right (Left) Waist and Fix of a Transformation, Idempotents

  JOURNAL NAME: Open Journal of Discrete Mathematics, Vol.11 No.3, July 13, 2021

  ABSTRACT: Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .