Cryptomorphism

In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent (possibly in some informal sense) but not obviously equivalent.

Etymology

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.

Use in matroid theory

Its informal sense was popularized (and greatly expanded in scope) by Gian-Carlo Rota in the context of matroid theory: there are dozens of equivalent axiomatic approaches to matroids, but two different systems of axioms often look very different.

In his 1997 book Indiscrete Thoughts, Rota describes the situation as follows:

Like many another great idea, matroid theory was invented by one of the great American pioneers, Hassler Whitney. 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 (possibly to Rota's disappointment). It is, however, in fairly wide use among researchers in matroid theory.

References

• Birkhoff, G. Latice Theory, 3rd edition. American Mathematical Society Colloquium Publications, Vol. XXV. 1967.

• Crapo, H. and Rota, G-C., On the foundations of combinatorial theory: Combinatorial geometries. M.I.T. Press, Cambridge, Mass. 1970.

• Rota, G-C. Indiscrete Thoughts, Birkhäuser Boston, Inc., Boston, MA. 1997.

• White, N. Editor. Theory of Matroids, Encyclopedia of Math and it's applications, 26. Cambridge University Press, Cambridge. 1986.