Examples of cryptomorphic definitions abound in matroid theory and others can be found elsewhere, e.g., in group theory the definition of a group by a single operation of division, which is not obviously equivalent to the usual three "operations" of identity element, inverse, and multiplication.
The word was coined by Garrett Birkhoff before 1967, for use in the third edition of his book Lattice Theory.
Birkhoff did not give it a formal definition, though others working in the field have made some attempts since.
His paper, which is still today the best entry to the subject, flagrantly reveals the unique peculiarity of this field, namely, the exceptional variety of cryptomorphic definitions for a matroid, embarrassingly unrelated to each other and exhibiting wholly different mathematical pedigrees.
It is as if one were to condense all trends of present day mathematics onto a single finite structure, a feat that anyone would a priori deem impossible, were it not for the fact that matroids do exist.Though there are many cryptomorphic concepts in mathematics outside of matroid theory and universal algebra, the word has not caught on among mathematicians generally.