In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant...

cone G+. If the order on the group is a linear order, then it is said to be a linearly ordered group. If the order on the group is a lattice order, i.e. any...

cyclic groups Z/n. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational...

ordered groups. One can also glue together different linearly ordered sets to form a circularly ordered set. For example, given two linearly ordered sets...

said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts...

zero. A group is a linearly ordered group when, in addition, its elements can be linearly ordered in a way that is compatible with the group operation:...

property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition. In an ordered ring, no negative element is a...

set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term...

Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order. Archimedean group: a linearly ordered group...

forms a fan. Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb Ordered group – Group with a compatible...

applies". (Some authors fix y to be zero, relying on the real number's linearly ordered group structure for addition. In classical logic, this axiom of trichotomy...

dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named...

stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group of real numbers, the...

elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning...

Quantale Partially ordered monoid Ordered group Archimedean property Ordered ring Ordered field Artinian ring Noetherian Linearly ordered group Monomial order...

the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function...

refer to: Linear order (linguistics), the order of words or phrases in linguistics Dense linear order, in mathematics Linearly ordered group This disambiguation...

(under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the ordering principle, OP...

x,y,z\in A.} Then: Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb Ordered field – Algebraic object...

are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent...

total order are comparable. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real...

Long line (topology) Linear continuum Order topology (functional analysis) Partially ordered space Lynn, I. L. (1962). "Linearly orderable spaces". Proceedings...

is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and...

wikidata descriptions as a fallback Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages...

also the theories of all the following structures: linear orders, trees, abelian linearly ordered groups, algebraically closed valued fields, and the p-adic...

linearly ordered. It was introduced by Mostowski (1939). The Mostowski model can be constructed as the permutation model corresponding to the group of...

order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations...

functional analysis are typically not linearly topologized (since subspaces do not form a neighborhood system). Ordered topological vector space Ring of restricted...

of all linearly independent subsets of V. Since V is not the zero vector space, there exists a nonzero element v of V, so P contains the linearly independent...

(the link function) to vary linearly with the predictors (rather than assuming that the response itself must vary linearly). For example, the case above...