  We propound the thesis that there is a limitation to the number of possible
structures which are axiomatically endowed with identities involving
operations. In the case of algebras with a binary operation satisfying a
formally reducible (to be explained) identity between two n-iterates of the
operation, it is established that the frequency of such algebras goes to zero
as n tends to infinity. This is proved by a suitable ordering and labeling of
the expressions (words) of the corresponding free algebra and the formation of
a series of tableaux whose entries are the labels. The tableaux reveal
surprising symmetry properties, stated in terms of the Catalan numbers and
their partitions. As a result of the defining identity and the tableaux all
iterates of order higher than n fall into equivalence classes of semantically
equal ones. Classnumbers depending on the tableaux are calculated for all
algebras of order n = 3 (and partially for n = 4). Certain classnumbers are
invariants in the sense that for algebras of same order they are equal.
Algebras with several operations of any arity are considered. A generalization
of Catalan numbers depending on homorphisms of the structure is proposed and
corresponding generating functions set up. As an example of this, a kind of
skein polynomials are constructed characterizing the formal build-up of
iterates. As no distinction is made between the various algebras, isomorphic or
not, which are models of the identity the results can also be formulated in
terms of varieties with signature the identity in question.
