Cuts in cyclically ordered sets
1984; Springer Nature; Volume: 34; Issue: 2 Linguagem: Inglês
10.21136/cmj.1984.101955
ISSN1572-9141
Autores Tópico(s)Advanced Numerical Analysis Techniques
ResumoAn ordered set is a pair (G, <) where G is a set and < is an order on G, i.e. an irreflexive and transitive binary relation on G.We write briefly G instead of (G, <) if the order < is given.If < is an order on G, then the dual relation <* = > is an order on G.An element x e G is called the least element of (G, <) iïï x < y for any y e G ~ {x}; the greatest element is defined dually.If (G, <) is an ordered set and H Я G, then < n Я^ is an order on Я; this order is denoted by <|д or, briefly, also <, and the subset H = (Я, <) is called an ordered subset of the ordered set G = (G, <).An order < on a set G is linear iff* x < }^ or j; < x for any x, y e G, X Ф j; in this case (G, <) is called a linearly ordered set.I.I.Definition.Let (G, <c), (Я, <^) be ordered sets with G n H = 0.An ordinal sum G ® H of ordered sets G, Я is the set G u Я with the binary relation < defined by x < у iïï either x, y e G, x <ду or x, yeH^x <ну or xeG, yeH.It is known ([!]; but it is trivial to prove it) that < is an order on G u Я so that G ® Я is an ordered set.Further, the operation @ is associative so that the symbol Gl © G2 Ф ... © G" is defined
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