This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Linearly disjoint" – news · newspapers · books · scholar · JSTOR(May 2024)
In mathematics, algebras A, B over a field k inside some field extension of k are said to be linearly disjoint over k if the following equivalent conditions are met:
(i) The map induced by is injective.
(ii) Any k-basis of A remains linearly independent over B.
(iii) If are k-bases for A, B, then the products are linearly independent over k.
Note that, since every subalgebra of is a domain, (i) implies is a domain (in particular reduced). Conversely if A and B are fields and either A or B is an algebraic extension of k and is a domain then it is a field and A and B are linearly disjoint. However, there are examples where is a domain but A and B are not linearly disjoint: for example, A = B = k(t), the field of rational functions over k.
One also has: A, B are linearly disjoint over k if and only if subfields of generated by , resp. are linearly disjoint over k. (cf. Tensor product of fields)
Suppose A, B are linearly disjoint over k. If , are subalgebras, then and are linearly disjoint over k. Conversely, if any finitely generated subalgebras of algebras A, B are linearly disjoint, then A, B are linearly disjoint (since the condition involves only finite sets of elements.)
some field extension Ω {\displaystyle \Omega } of k are said to be linearlydisjoint over k if the following equivalent conditions are met: (i) The map...
and Y {\displaystyle Y} are T {\displaystyle T} -linearlydisjoint if and only if for all linearly independent sequences x 1 , … , x m {\displaystyle...
theory.) Subfields K and L of M are linearlydisjoint (over a subfield N) when in this way the natural N-linear map of K ⊗ N L {\displaystyle K\otimes...
{\displaystyle x} . Equivalently, two sets are linearly separable precisely when their respective convex hulls are disjoint (colloquially, do not overlap). In simple...
Extension of scalars Tensor product of modules Tensor product of fields Linearlydisjoint Multilinear subspace learning Kassel (1995), p. 32. Lang 2002, pp...
a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term chain is sometimes defined...
k} (that is, to say, L , k ¯ {\displaystyle L,{\overline {k}}} are linearlydisjoint over k). Regularity is transitive: if F/E and E/K are regular then...
is a separable extension of F, E p {\displaystyle E^{p}} and F are linearlydisjoint over F p , {\displaystyle F^{p},} F 1 / p ⊗ F E {\displaystyle F^{1/p}\otimes...
closure of F in E is purely inseparable over F; equivalently, E is linearlydisjoint from the separable closure of F. Purely transcendental extension An...
and Y {\displaystyle Y} are T {\displaystyle T} -linearlydisjoint if and only if for all linearly independent sequences x 1 , … , x m {\displaystyle...
of any composition series. linearlydisjoint Two subfields of a field extension K over a field k are called linearlydisjoint if the natural map from their...
k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearlydisjoint. André Weil...
level of function fields it is possible to construct J by taking linearlydisjoint copies of the function field of C, and within their compositum taking...
the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory...
\gamma )} . The latter part is usually ignored under the assumption of disjoint parameter sets. More so, under classic assumptions X {\displaystyle \mathbf...
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...
the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an...
inseparable over K. An extension L/K is primary if and only if it is linearlydisjoint from the separable closure of K over K. A subextension of a primary...
open intervals and rays. A topological space X is called orderable or linearly orderable if there exists a total order on its elements such that the order...
topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological...
structures has the strong amalgamation property (SAP), also called the disjoint amalgamation property (DAP), if for every amalgam with A,B,C ∈ K there...
connected graph is Eulerian if and only if it is the disjoint union of some cycles. Hierholzer developed a linear time algorithm for constructing an Eulerian tour...
K-ary tree And–or tree (a,b)-tree Link/cut tree SPQR-tree Spaghetti stack Disjoint-set data structure (Union-find data structure) Fusion tree Enfilade Exponential...
In computational geometry, a maximum disjoint set (MDS) is a largest set of non-overlapping geometric shapes selected from a given set of candidate shapes...
order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S of more than...
O(n)} . Informally, this means that the running time increases at most linearly with the size of the input. More precisely, this means that there is a...
}}y} in a vector lattice X {\displaystyle X} are said to be lattice disjoint or disjoint if inf { | x | , | y | } = 0 , {\displaystyle \inf\{|x|,|y|\}=0,}...