Separation relation

In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation S(a, b, c, d) satisfying certain axioms, which is interpreted as asserting that a and c separate b from d.^[1]

Whereas a linear order endows a set with a positive end and a negative end, a separation relation forgets not only which end is which, but also where the ends are located. In this way it is a final, further weakening of the concepts of a betweenness relation and a cyclic order. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivial reducts of the ordered set of rational numbers.^[2]

^ Huntington, Edward V. (July 1935), "Inter-Relations Among the Four Principal Types of Order" (PDF), Transactions of the American Mathematical Society, 38 (1): 1–9, doi:10.1090/S0002-9947-1935-1501800-1, retrieved 8 May 2011
^ Macpherson, H. Dugald (2011), "A survey of homogeneous structures" (PDF), Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, retrieved 28 April 2011

[1] Huntington, Edward V. (July 1935), "Inter-Relations Among the Four Principal Types of Order" (PDF), Transactions of the American Mathematical Society, 38 (1): 1–9, doi:10.1090/S0002-9947-1935-1501800-1, retrieved 8 May 2011

[2] Macpherson, H. Dugald (2011), "A survey of homogeneous structures" (PDF), Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, retrieved 28 April 2011

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Separation relation

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