The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.
Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication.
All that can be done here is to give a sketch of the more important and independent special algebras at present known to exist.
The types of linear associative algebras, not assumed to be commutative, have been enumerated (with some omissions) up to sextuple algebras inclusive by B.
Various special algebras (for example, quaternions) may be expressed in the notation of the algebra of matrices.
This applies also to quaternions, but not to extensive quantities, nor is it true for linear algebras in general.
The algebras discussed up to this point may be considered as independent in the sense that each of them deals with a class of symbols of quantity more or less homogeneous, and a set of operations applying to them all.
In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of algebra (at present a hopeless task), they would have to be taken into account.
In recent times many mathematicians have formulated other kinds of algebras, in which the operators do not obey the laws of ordinary algebra.
Combebiac, Calcul des triquaternions (Paris, 1902); Don Francisco Perez de Munoz, Introduction al estudio del cdlculo de Cuaterniones y otras Algebras especiales (Madrid, 1905); A.