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Separability may refer to: Mathematics. Separable algebra, a generalization to associative algebras of the notion of a separable field extension;
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane ) by thinking of one set of points as being colored blue and the other set of points as being colored red.
Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.
In field theory, a branch of algebra, an algebraic field extension / is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). [1]
The separability problem is a subject of current research. A separability criterion is a necessary condition a state must satisfy to be separable. In the low-dimensional ( 2 X 2 and 2 X 3 ) cases, the Peres-Horodecki criterion is actually a necessary and sufficient condition for separability.
Kirchberger's theorem is a theorem in discrete geometry, on linear separability.The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the ...
In the multipartite case there is no simple necessary and sufficient condition for separability like the one given by the PPT criterion for the and cases. However, many separability criteria used in the bipartite setting can be generalized to the multipartite case.
The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand in its restricted, induced module. But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables , which induce to a finite number of constituent indecomposable ...