## Associativity of the tensor product

All vector spaces are by default over or . We consider three vector spaces , and . Let be a tensor product of and , and a tensor product of and . The aim is first to show that , together with the mapping:     is a tensor product of , and . Proof:…

## Free associative algebra on a vector space

Vector spaces and associative algebras will be implicitely all on the same field or . Let be a vector space. We wish to build a free associative algebra on . For any natural number , we will note (tensor product times). Each is a vector space. We may form:     An element of can…

## Lower and upper limits of a filter base

Two cases, that I have been unable to unify: a filter base over a power set with a set, or over . First case: filter bases over a power set filter base over , with a set. Definitions         Property It is always the case that . Proof: Let be an eventual…

## Intersection of filters

Let and be filters on a same set . Then is also a filter on . Proof: belongs to both and , hence it belongs to , and . Since is not an element of , it is not an element of . Let be an element of and a superset of . Then and…

## Concrete categories: injective/surjective implies mono/epi; functor generalization

A category is by definition concrete if there exists a faithful functor from to the category of sets . (A functor is faithful if it is one-to-one (injective) regarding morphisms.) It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections. We…

## Degree and Euclidean division of polynomials

For the definition and characterization of the polynomials over a field , see Polynomials over a field. Definition of the degree of a polynomial The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers….

## Polynomials over a field

In this context, we consider a commutative field (simply: field) . The polynomials will be constructed as a certain associative unital algebra over (“-aua”), together with a distinguished element called the formal variable. itself, as a vector space over itself and the usual multiplication in as the third law, is a -aua, Morphisms between two…

## Definitions concerning filters

Definition of a filter Let be a set. Traditional definition of a filter on : A collection of subsets of is a filter on iff: is non-empty (equivalently in context: ); ; ; . It follows from the definition that there can be no filters on the empty set. If we waive the second rule…