Conjugacy classes of the order-preserving and order-decreasing partial one-to-one transformation semigroups

Abstract

In 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.