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Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement, as in cutting a deck of cards. [3] One may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above. [4] Cutting a single point out of a cyclic order leaves a linear order behind.
By analogy with an Archimedean linearly ordered group, one can define an Archimedean cyclically ordered group as a group that does not contain any pair of elements x, y such that [e, x n, y] for every positive integer n. [3] Since only positive n are considered, this is a stronger condition than its linear counterpart.
Cyclic number, a number such that cyclic permutations of the digits are successive multiples of the number; Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
Cycle, a set equipped with a cyclic order. Necklace (combinatorics), an equivalence classes of cyclically ordered sequences of symbols modulo certain symmetries; Cyclic (mathematics), a list of mathematics articles with "cyclic" in the title; Cyclic group, a group generated by a single element
A discrete example is a finite cyclic group of order n. ... Convolution and related operations are found in many applications in science, engineering and mathematics.
The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication , the order of an element a of a group, is thus the smallest positive integer m such that a m = e , where e denotes the identity element of the group, and a m ...
This relation is a partial cyclic order, but it cannot be extended with either abc or cba; either attempt would result in a contradiction. [4] The above was a relatively mild example. One can also construct partial cyclic orders with higher-order obstructions such that, for example, any 15 triples can be added but the 16th cannot.