Cryptomorphism
Encyclopedia
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. This word is a play on the many morphism
s in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism
", "homomorphism
", or "morphisms".
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.
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:
Though there are many cryptomorphic concepts in mathematics outside of matroid theory and universal algebra
, the word has not caught on among mathematicians generally. It is, however, in fairly wide use among researchers in matroid theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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. This word is a play on the many morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
", "homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
", or "morphisms".
Etymology
The word was coined by Garrett BirkhoffGarrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
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 RotaGian-Carlo Rota
Gian-Carlo Rota was an Italian-born American mathematician and philosopher.-Life:Rota was born in Vigevano, Italy...
in the context of matroid
Matroid
In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....
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:
Though there are many cryptomorphic concepts in mathematics outside of matroid theory and universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, the word has not caught on among mathematicians generally. It is, however, in fairly wide use among researchers in matroid theory.