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It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partial cyclic order that satisfies an additional spreading axiom. [29] Replacing the asymmetry axiom with a complementary version results in the definition of a co-cyclic order.
Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group T and its subgroups , such as the subgroup of rational points .
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
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
Alternation of generations – Beta oxidation – Bioelectricity – Biological pest control – Biological rhythm – Bipolar disorder – Cardiopulmonary resuscitation – Calvin–Benson cycle – Cell cycle – Chronobiology – Citric acid cycle – Circadian rhythm – Clinical depression – Digestion – Ecology – Feedback – Infradian rhythm - Life cycle – List of biochemistry ...
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 ...
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with a k = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows
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.