ordered group

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• Ordered group — In abstract algebra, an ordered group is a group (G,+) equipped with a partial order ≤ which is translation invariant ; in other words, ≤ has the property that, for all a , b , and g in G , if a ≤ b then a+g ≤ b+g and g+a ≤ g+b . Note that… …   Wikipedia

• Cyclically ordered group — In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger… …   Wikipedia

• Partially-ordered group — In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order ≤ that is translation invariant; in other words, ≤ has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b. An element x… …   Wikipedia

• Linearly ordered group — In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation le; is total. This means that the following statements hold for all a , b , c isin; G :* if a le; b and b le; a then a = b… …   Wikipedia

• Ordered semigroup — In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered… …   Wikipedia

• Cyclic group — Group theory Group theory …   Wikipedia

• Ordered pair — In mathematics, an ordered pair (a, b) is a pair of mathematical objects. In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Alternatively, the objects are called the first and… …   Wikipedia

• Group tournament ranking system — In a group tournament, unlike a knockout tournament, there is no decisive final match. Instead, all the competitors are ranked by examining the results of all the matches played in the tournament. Points are awarded for each fixture, with… …   Wikipedia

• ordered — É”rdÉ™(r)d / ɔː adj. well arranged, neat; commanded, required or·der || É”rdÉ™(r) / ɔːd n. arrangement; instruction; command; request for something; religious group v. command; request something; arrange; manage …   English contemporary dictionary

• Archimedean group — In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure consisting of a set together with a binary operation and binary relation satisfying certain axioms detailed below. We can also say that an Archimedean… …   Wikipedia

• Partially ordered set — The Hasse diagram of the set of all subsets of a three element set {x, y, z}, ordered by inclusion. In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering,… …   Wikipedia