Purely Idempotent Latin Squares And Purely Symmetric Idempotent Latin Squares

Combinatorial design theory is an important branch of modern combinatorial theory. The research of design refers to a very important and central problem of combinatorial theory. That is to say we must arrange some things by certain rule. In this article, we completely solved the existence problem of purely idempotent Latin squares and purely symmetric idempotent Latin squares.In Chapter 1, introduce some basic conceptions and theorems.In Chapter 2, discuss the existence of purely idempotent Latin square and list some constructions and results. Give a complete solution of purely idempotent Latin squares. Namely purely idempotent Latin square of order n exists if and only if n ≥ 8.In Chapter 3, discuss the existence of purely symmetric idempotent Latin squares and list some constructions and results. Give a complete solution of purely idempotent Latin squares. Namely purely symmetric idempotent Latin square of order n exists if and only if n ≥ 5.