The Fine Structures Of Three Idempotent Latin Squares

In1991, Colbourn et al. determined the fine structure of threefold triple systems. In2002, Adams et al. completely solved the so called3-way intersection problem: determining, for all orders v, the set I3[v] of integers k for which there exist three Latin squares of orders v with the fine structure (0, v2-k). Denote by Fin(v) the set of all integer pairs (t, s) for which there exist three Latin squares of orders v on the same set having fine structure (t, s). In2006, Y. Chang, G. Lo Faro, and G. Nordo determined the set Fin(v) for any integer v≥10. In2008, Z. Wang and Y. Chang determined the set Fin(8), and some results on Fin(v) with5≤v≤7were updated. Denote by SFin(v) the set of all integer pairs(t, s) for which there exist three symmetric Latin squares of orders v on the same set having fine structure (t, s). In2008, E. Fen and Y. Chang determined the set SFin(2n) for any integer n≥5.Denote by IdFin(v) the set of all integer pairs (t, s) for which there exist three idempotent Latin squares of orders v on the same set having fine structure (t, s). In this thesis, some results on IdFin(v) are obtained for any integers v∈[7,10]; and the elements (6,17),(4,18) in IdFin(7)(?) Fin(7) are found, which updates the previous results. And it is proven that (0,9)(?) IdFin(v) for any integers v (?)[5,8], and (t,16)(?) IdFin(v) for any integers v∈[6,11] and t∈[0,3]. In this thesis, the method of recursive constructing idempotent Latin squares of higher order with fine structure is given, and the set IdFin(v) for any integer v with v(?)[7,10] is completely determined.This thesis is divided into eight chapters.In the first chapter, we introduce the basic definition, some known results and recent developments in the study of Latin Squares. In the second chapter, we present some lemmas, construct fine structures for certain small order idempotent Latin squares, and show that they do not exist for some small order cases. In the third chapter, we present an important method of recursive constructions. In the forth chapter, we construct some specific elements of IdFin(v) and completely determine the set IdFin(v) for any integer v with v∈[11,24]. In the fifth chapter, we give another method of constructing elements of IdFin(v). In the sixth chapter, we discuss non-semistable components containing oriented cycles. In the seventh chapter, some application of grey clustering assessment is given. In the eighth chapter, we state and prove the important result IdFin(v)=IdAdm(v) for any integer v≥12, and summarize the main results. And some problems which will be further studied are put forward.