The Fine Structures Of Three Symmetric Latin Squares

Let L_k = (a_ij^k)_v×v, k = 1,2,3 be symmetric Latin squares on symbol set 5 containingv elements. DefineThe fine structure of 3 symmetric Latin squares L₁,L₂,L₃ is the vector (t, s), where t is the number of elements in C₂ and s is the number of elements in C₂∪C₃. Let S Fin(v) denote the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v with fine structure (t, s).Definewhere A = {(t,t) : t∈[4,11] \ {5}}∪{(6,7), (6,8), (0,9), (6,9), (8,9), (8,10), (9,10), (6,11), (8,11), (9,11), (10,11)}, andε= {(1,12), (2,12), (5,12), (0,13), (1,13), (2,13), (3,13), (5,13), (0,14), (1,14), (1,15), (1,17)}.LetIn this thesis, we completely determine the set S Fin(2n) for any integer n≥5. Some results on S Fin(v) with v = 6,7,8, and some recursive constructions are given.The first chapter introduces the basic definition and some known results. In the secondchapter, we present some lemmas. Then, we present some recursive constructions and examples for small orders. At last, we determine the set S Fin(2n) with 5≤n≤9. In the third chapter, we summarize the results and give S Fin(2v) = S Adm(2v) for any integer v≥5.