Notes On Usage
^cyclic order The relation may be called a cyclic order (Huntington 1916, p. 630), a circular order (Huntington 1916, p. 630), a cyclic ordering (Kok 1973, p. 6), or a circular ordering (Mosher 1996, p. 109). Some authors call such an ordering a total cyclic order (Isli & Cohn 1998, p. 643), a complete cyclic order (Novák 1982, p. 462), a linear cyclic order (Novák 1984, p. 323), or an l-cyclic order or ℓ-cyclic order (Černák 2001, p. 32), to distinguish from the broader class of partial cyclic orders, which they call simply cyclic orders. Finally, some authors may take cyclic order to mean an unoriented quaternary separation relation (Bowditch 1998, p. 155).
^cycle A set with a cyclic order may be called a cycle (Novák 1982, p. 462) or a circle (Giraudet & Holland 2002, p. 1). The above variations also appear in adjective form: cyclically ordered set (cyklicky uspořádané množiny, Čech 1936, p. 23), circularly ordered set, total cyclically ordered set, complete cyclically ordered set, linearly cyclically ordered set, l-cyclically ordered set, ℓ-cyclically ordered set. All authors agree that a cycle is totally ordered.
^ternary relation There are a few different symbols in use for a cyclic relation. Huntington (1916, p. 630) uses concatenation: ABC. Čech (1936, p. 23) and (Novák 1982, p. 462) use ordered triples and the set membership symbol: (a, b, c) ∈ C. Megiddo (1976, p. 274) uses concatenation and set membership: abc ∈ C, understanding abc as a cyclically ordered triple. The literature on groups, such as Świerczkowski (1959a, p. 162) and Černák & Jakubík (1987, p. 157), tend to use square brackets: . Giraudet & Holland (2002, p. 1) use round parentheses: (a, b, c), reserving square brackets for a betweenness relation. Campero-Arena & Truss (2009, p. 1) use a function-style notation: R(a, b, c). Rieger (1947), cited after Pecinová 2008, p. 82) uses a "less-than" symbol as a delimiter: < x, y, z <. Some authors use infix notation: a < b < c, with the understanding that this does not carry the usual meaning of a < b and b < c for some binary relation < (Černy 1978, p. 262). Weinstein (1996, p. 81) emphasizes the cyclic nature by repeating an element: p ↪ r ↪ q ↪ p.
^embedding Novák (1984, p. 332) calls an embedding an "isomorphic embedding".
^roll In this case, Giraudet & Holland (2002, p. 2) write that K is L "rolled up".
^orbit space The map T is called archimedean by Bowditch (2004, p. 33), coterminal by Campero-Arena & Truss (2009, p. 582), and a translation by McMullen (2009, p. 10).
^universal cover McMullen (2009, p. 10) calls Z × K the "universal cover" of K. Giraudet & Holland (2002, p. 3) write that K is Z × K "coiled". Freudenthal & Bauer (1974, p. 10) call Z × K the "∞-times covering" of K. Often this construction is written as the anti-lexicographic order on K × Z.
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