An isomorphism is a one to one mapping of the elements of one set onto another such that the result of an operation on the elements of one set are identical to the result of the same operation on the elements of the other set. An isomorphism is also known as bijective morphism, a mapping of one set onto another set without the loss of information. Isomorphism is derived from the two Greek words iso and morphism. The Greek word iso means equal or identical. The Greek word morphism is actually composed of two other Greek words, morph meaning form or shape and ism indicating an action. Combining these two words together into morphism yields a word meaning to form or shape. The symbol indicating isomorphic behavior is usually denoted as ( and is written A ( B, meaning A is isomorphic to B. This symbol is also sometimes used to mean geometric congruence in mathematics.
Isomorphism was first discovered by German scientist Eilhard Mitscherlich in 1819. Mitscherlich observed that chemical compounds having the same number of atoms per molecule are predisposed to form crystals that have identical angles. He called this property isomorphism and continued to advance the theory of isomorphism, which describes a relationship between crystalline structure and chemical composition. After his work was verified it became one of the basic levers of the atomic theory.
The concept of isomorphism was extended to mathematics to mean two different systems that are the same but does not mean that they are equal. For example, for two numerical systems to have the same structure, each system must contain a number that has a counterpart in the other system. For the two systems to be isomorphic: 1) there is a mapping of one system to the other that put them into a one to one correspondence and 2) in this mapping, the results of mathematical operations, such as addition and multiplication, are preserved. An example of two number systems that are isomorphic to one another are Arabic and Roman.These systems use different symbols to represent numbers but the results of addition and multiplication are equal in the two systems.
The identities and inverses of a group are preserved in an isomorphism. Isomorphism can be applied to spaces, shapes and groups. A vector space in which addition and scalar multiplication are preserved is a space isomorphism. There are some specific isomorphisms that have unique names describing them. An automorphism is the result of an isomorphism of a group onto itself. Mapping a geometric figure or topological space onto another figure or space is called a homeomorphism. This equivalence relation is continuous in both directions and as such is also known as a continuous transformation. Isometry is a type of homeomorphism in which distances are also preserved. Although all of these types of transformation are specific to the details of each situation they are all still isomorphisms.
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