Mathematical Isomorphic Objects: Definition and Basic Application
Isomorphic is a term derived from the concept of isomorphism which is a common concept in the field of mathematics. This comes from a Greek word “iso” that means equal and “morphosis” which means “to shape” or “to form”.
Basic Concept of Isomorphic Objects
In the field of mathematics, two objects are said to be isomorphic if isomorphism is present between them. A similar concept is that of automorophism is related to isomorphism where the target and source match. The idea of isomorphism is present with the fact that two of such subjects cannot be differentiated through the use of characteristics that are used for defining morphisms, therefore such objects might be taken as similar as far as one takes into account only such characteristics and their results.
Application of Isomorphic Objects in Abstract Algebra
There are two basic kinds of isomorphisms in the field of abstract algebra:
This is the type of isomorphism which is present among groups.
This kind of isomorphism exists between rings. It should be noted that the isomorphisms that are present among fields are basically the ring isomorphism.
Similar to the way the isomorphisms of algebraic framework develop group, the isomorphism among two algebras which have a mutual framework develop a heap. Allowing a specific isomorphism recognize the two frameworks converts the heap to a group.
Isomorphism and Mathematical Analysis
With respect to mathematical analysis, the term of Laplace is used for the transformation of isomorphic mapping differential sums to simpler algebraic sums. Similarly, in the same kind of analysis, isomorphisms which exists between two Hilbert spaces is basically a bijection showing addition, inner product as well as scalar multiplication.
Logical Atomism and Isomorphism
In the initial theories related to logical atomism, the real link between true proposition and facts was developed by Ludwig Wittgenstein and Bertrand Russell to be having isomorphism. Instances of such measures can be seen in the “Introduction to Mathematical Philosophy” by Bertrand Russell.