A basis (or reference frame) of a (universal) algebra is a function
, that takes algebra elements as values (even outside the basis set) will be denoted by
, 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
and the numbers of elements in the arguments, or “arity”, of the operations are finite, this is the finitary multiple composition .)
that takes every (sample) m as argument to extend it onto an endomorphism
[5] Several other conditions that characterize bases for universal algebras are omitted.
Yet, contrary to the vector space case, a universal algebra might lack bases and, when it has them, their dimension sets might have different finite positive cardinalities.
[6] In the universal algebra corresponding to a vector space with finite dimension the bases essentially are the ordered bases of this vector space.
The functions m for the inner condition correspond to the square arrays of field elements (namely, usual vector-space square matrices) that serve to build the endomorphisms of vector spaces (namely, linear maps into themselves).
Then, the inner condition requires a bijection property from endomorphisms also to arrays.
In fact, each column of such an array represents a vector
For instance, when the vectors are n-tuples of numbers from the underlying field and b is the Kronecker basis, m is such an array seen by columns,
is the sample of such a linear map at the reference vectors and
When the vector space is not finite-dimensional, further distinctions are needed.
formally have an infinity of vectors in every argument, the linear combinations they evaluate never require infinitely many addenda
, both the linear independence and spanning properties for infinite basis sets follow from present outer condition and conversely.
Therefore, as far as vector spaces of a positive dimension are concerned, the only difference between present bases for universal algebras and the ordered bases of vector spaces is that here no order on
When the space is zero-dimensional, its ordered basis is empty.
Yet, since this space only contains the null vector and its only endomorphism is the identity, any function b from any set
(even a nonempty one) to this singleton space works as a present basis.
This is not so strange from the point of view of universal algebra, where singleton algebras, which are called "trivial", enjoy a lot of other seemingly strange properties.
be an "alphabet", namely a (usually finite) set of objects called "letters".
in case of the empty word (formal language notation).
Then, the inner condition will immediately prove that one of its bases is the function b that makes a single-letter word
(Depending on the set-theoretical implementation of sequences, b may not be an identity function, namely
[7]) In fact, in the theory of D0L systems (Rozemberg & Salomaa 1980) such
are the tables of "productions", which such systems use to define the simultaneous substitutions of every
is the well-known bijection that identifies every word endomorphism with any such table.
(The repeated applications of such an endomorphism starting from a given "seed" word are able to model many growth processes, where words and concatenation serve to build fairly heterogeneous structures as in L-system, not just "sequences".)