In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the ''same'' object are "cryptomorphic" if it is not obvious that they define the same object. Examples of cryptomorphic definitions abound in
matroid theory and others can be found elsewhere, e.g., in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
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.
This word is a play on the many
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s in mathematics, but "cryptomorphism" is only very distantly related to "
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
", "
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
", or "morphisms". The equivalence may in a cryptomorphism, if it is not actual identity, be informal, or may be formalized in terms of a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
or
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
between the mathematical objects defined by the two cryptomorphic axiom systems.
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:
{{cquote, 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
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
, the word has not caught on among mathematicians generally. It is, however, in fairly wide use among researchers in matroid theory.
See also
*
Combinatorial class In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size.
Counting sequences and isomorphis ...
, an equivalence among
combinatorial enumeration problems hinting at the existence of a cryptomorphism
References
* Birkhoff, G.: ''Lattice 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.
* Elkins, James: Chapter ''Cryptomorphs'' in ''Why Are Our Pictures Puzzles?: On the Modern Origins of Pictorial Complexity'', 1999
* Rota, G.-C.: ''Indiscrete Thoughts'', Birkhäuser Boston, Inc., Boston, MA. 1997.
* White, N., editor: ''Theory of Matroids'', Encyclopedia of Mathematics and its Applications, 26. Cambridge University Press, Cambridge. 1986.
Mathematical terminology
Matroid theory