Basis (universal algebra)

In universal algebra, a basis is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner. It also represents the endomorphisms of an algebra by certain indexings of algebra elements, which can correspond to the usual matrices when the free algebra is a vector space. A basis (or reference frame) of a (universal) algebra is a function that takes some algebra elements as values and satisfies either one of the following two equivalent conditions. Here, the set of all is called the basis set, whereas several authors call it the "basis". The set of its arguments is called the dimension set. Any function, with all its arguments in the whole , that takes algebra elements as values (even outside the basis set) will be denoted by . Then, will be an . This condition will define bases by the set of the -ary elementary functions of the algebra, which are certain functions that take every as argument to get some algebra element as value In fact, they consist of all the projections with in which are the functions such that for each , and of all functions that rise from them by repeated "multiple compositions" with operations of the algebra. (When an algebra operation has a single algebra element as argument, the value of such a composed function is the one that the operation takes from the value of a single previously computed -ary function as in composition. When it does not, such compositions require that many (or none for a nullary operation) -ary functions are evaluated before the algebra operation: one for each possible algebra element in that argument. In case and the numbers of elements in the arguments, or “arity”, of the operations are finite, this is the finitary multiple composition .) Then, according to the outer condition a basis has to generate the algebra (namely when ranges over the whole , gets every algebra element) and must be independent (namely whenever any two -ary elementary functions coincide at , they will do everywhere: implies ).