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The free product G ∗ H is the group whose elements are the reduced words in G and H, under the operation of concatenation followed by reduction. For example, if G is the infinite cyclic group x {\displaystyle \langle x\rangle } , and H is the infinite cyclic group y {\displaystyle \langle y\rangle } , then every element of G ∗ H is an ...
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
For example, a polynomial of degree n has a pole of degree n at infinity. The complex plane extended by a point at infinity is called the Riemann sphere. If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its ...
For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces ), the coproduct, called the direct sum , consists of the elements of the direct product which have only finitely many nonzero terms.
The six independent scalar products g ij =h i.h j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g ij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g 11 =h 1 h 1, g 22 =h 2 h 2, g 33 =h 3 h 3.
Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. [1] The biproduct is a generalization of finite direct sums of modules.
A subdirect product is a subalgebra (in the sense of universal algebra) A of a direct product Π i A i such that every induced projection (the composite p j s: A → A j of a projection p j: Π i A i → A j with the subalgebra inclusion s: A → Π i A i) is surjective. A direct (subdirect) representation of an algebra A is a direct (subdirect ...