| Purely tetrahedral quadruple systems |
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| 引用本文： | Jl Lijun Department of Mathematics,Suzhou University,Suzhou 215006,China. Purely tetrahedral quadruple systems[J]. 中国科学A辑(英文版), 2006, 49(10): 1327-1340. DOI: 10.1007/s11425-006-2013-5 |
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| 作者姓名： | Jl Lijun Department of Mathematics Suzhou University Suzhou 215006 China |
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| 作者单位： | Jl Lijun Department of Mathematics，Suzhou University，Suzhou 215006，China |
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| 基金项目： | 国家自然科学基金;江苏省自然科学基金 |
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| 摘 要： | An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X,B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n = 2,4 (mod 6), n > 4, or n = 1,5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n = 2,4 (mod 6) and n > 4, or n = 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n=1,3 (mod 6), n > 3, or n =0,4 (mod 12).
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| 收稿时间： | 2006-04-04 |
| 修稿时间： | 2006-05-14 |
Purely tetrahedral quadruple systems |
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| JI Lijun. Purely tetrahedral quadruple systems[J]. Science in China(Mathematics), 2006, 49(10): 1327-1340. DOI: 10.1007/s11425-006-2013-5 |
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| Authors: | JI Lijun |
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| Affiliation: | Department of Mathematics, Suzhou University, Suzhou 215006, China |
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| Abstract: | An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X,B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n = 2,4 (mod 6), n > 4, or n = 1,5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n = 2,4 (mod 6) and n > 4, or n = 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n=1,3 (mod 6), n > 3, or n =0,4 (mod 12). |
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| Keywords: | large set t-wise balanced design Mendelsohn triple system quadruple system |
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