A tournament is an oriented graph where every pair of distinct vertices are adjacent. It is used as a result of various competitions, such as sports league result and voting. And as ranking, there are several linear orderings at the tournament such as ‘Slater order’, ‘Copeland order’, ‘Sorted sequence of kings’. And competition winner is the first vertex of linear order. We will focus on ‘Slater order’ and ‘Slater winner’. ‘Slater order’ is a linear order (v1, v2, ..., vn) of its vertex set such that | {(vi, vj ) : i < j} | is as large as possible. In this seminar, We will introduce several propositions for ‘Slater order’ and ’Slater winner’ and compare with other linear orders. And also introduce some methods for computing ‘Slater order’ of the given tournament by using boundary value of ‘Slater index’. Finally, as an example, we will calculate the ‘Slater order’ for a regular tournament, Brualdi-Li tournament and raise the question about the uniqueness of ‘Slater order’.