Olivine, an isomorphic series of forsterite and fayalite
The notion of isomorphism is used in the field of geology and in the field of mathematics . In the first case, the term refers to the property of that which is isomorphic .
Same crystal structure
Two bodies are isomorphic when, despite having different chemical compositions , they have the same crystalline structure and are able to associate to crystallize . This phenomenon is called isomorphism.
Broadly speaking, it can be said that isomorphism means that two different substances can jointly develop a single crystalline lattice . The group of possible mixtures is called isomorphic series : in them, one element can be replaced with another of similar size without any alterations in the crystalline structure.
If the charges presented by the exchanged atoms are not equal, we speak of heterovalent isomorphism and a substitution mechanism is required for compensation. On the other hand, if both atoms have the same charge, the isomorphism is isovalent .
The opposite of isomorphism is polymorphism . While isomorphic minerals have different chemical compositions and crystallize the same, the opposite occurs with polymorphic minerals.
One-to-one correspondence
For mathematics , on the other hand, isomorphism is the one-to-one correspondence that is registered between two algebraic structures, maintaining the operations. In this way, if there is isomorphism, the study of one structure can be reduced to that of the other.
Isomorphism is registered between two structures, in short, if each element of one corresponds only to one element of the other and the same occurs with each operation , also reciprocally. We must add that mathematical isomorphism is a morphism , that is, it is an application that does not alter the internal structure; more precisely, it is a homomorphism , because both objects have the same algebraic structure. So, given the definition of isomorphism, this morphism is of a specific type that gives rise to an inverse.
Continuing with this idea, it is also possible to define this concept as a bijective homomorphism , since its inverse function also falls into the category of homomorphism. The term "bijective" refers to the aforementioned correspondence of different elements in the departure and arrival sets in both directions . If two ordered sets present isomorphism, then they should be considered isomorphic and recognized as sets similar to each other . In the special case that both are equal, then one must speak of automorphism .
Before continuing we will talk about order sets : they are those in which their elements are related to each other by different operations, such as multiplication, and among which it is possible to recognize a minimum (from which the others start) and several maximals ( the maximums of each order branch). A completely ordered set, on the other hand, is one that relates all its elements, such that there is a single minimum and a single maximum . In one of them the only possibility of automorphism is the identity function , that is, the one that gives us its own argument as a result: the identity function for x = 4 is 4, and so on.
The isomorphism is a bijective homomorphism.
Equivalence relationship
In algebra and set theory, an equivalence relation is one that gives us the possibility of relating the elements of a set based on a given property or characteristic. Isomorphism also gives rise to this description for the following two reasons:
* is reflexive : the groups are isomorphic to themselves through the identity function, as we mentioned above. In addition to being a case of homomorphism and bijection;
* is symmetric : this is true because if the sets A and B are isomorphic, it is correct to describe such a relationship in both senses.