Ugbene, I. J.,Bakare, G. N.,Ibrahim G. R.2020-02-032020-02-032019-03Ugbene, I. J., Bakare, G. N., and Ibrahim G. R. (2019): Conjugacy classes of the order-preserving and order-decreasing partial one-to-one transformation semigroups. Journal of Science, Technology, Mathematics and Education.http://hdl.handle.net/123456789/3624In this paper I_n is considered to be the partial one-to-one transformation semigroup on . The order-preserving and order-decreasing partial one-to-one transformation semigroup(〖IC〗_n) is defined to be the subsemigroups of I_n. The nilpotent and idempotent elements of 〖IC〗_n were then obtained from its conjugacy classes using path structure to be n+1 and 1/192 n^4-1/16 n^3+29/48 n^2-n+2 respectively. The index and period of each conjugacy classes were also found. More so, some properties of conjugacy classes were stated.enSemigroup,Order-preservingNilpotentIdempotent Elements.Transformation Semigroup,Path Structure,Conjugacy,Order-decreasingPartial one-to-oneArticle