Isomorphism

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In mathematics, an isomorphism (from the Greek iso-morphos: Equal form) is a homomorphism (or more generally a morphism) that admits an inverse. The mathematical term for isomorphism is intended to capture the idea of having the same structure. Two mathematical structures between which an isomorphic relationship exists are called isomorphic.

Formal definition

It can be concisely defined as a bijective homomorphism such that its inverse is also a homomorphism. That is:

A isomorphism between two sets sorted ${displaystyle (P,leq)}$ and ${displaystyle (Q,leq ')}$ It's a bi-ective function. ${displaystyle {begin{array}{rrcl}h:Pto Q\end{array}}}$ such that:
For everything ${displaystyle p_{1},p_{2}in P}$ You have to ${displaystyle p_{1}leq p_{2}}$ Yes and only if ${displaystyle h(p_{1})leq 'h(p_{2})}$ .

If there is an isomorphism between ${displaystyle (P,leq)}$ and ${displaystyle (Q,leq ')}$ , then ${displaystyle (P,leq)}$ and ${displaystyle (Q,leq ')}$ It's called isomorphos and bijection $h$ is known as isomorphism between ${displaystyle (P,leq)}$ and ${displaystyle (Q,leq ')}$ . Plus, $P$ and $Q$ It's called similar each other.

Yeah. ${displaystyle P=Q}$ it is said that isomorphism is a automorphism. It can be shown that given a well-ordered set the only possible automorphism is the identity function.

Properties on Total Orders

Isomorphisms in linearly ordered sets have an equivalence relation, that is, they satisfy reflexivity, symmetry and transitivity, that is:

Sean. ${displaystyle (A,leq)}$ , ${displaystyle (B,leq ')}$ and ${displaystyle (C,leq '')}$ Linearly arranged sets, then:
${displaystyle (A,leq)}$ isomorph to ${displaystyle (A,leq)}$ .
Yeah. ${displaystyle (A,leq)}$ isomorph to ${displaystyle (B,leq ')}$ , then ${displaystyle (B,leq ')}$ isomorph to ${displaystyle (A,leq)}$ .
Yeah. ${displaystyle (A,leq)}$ isomorph to ${displaystyle (B,leq ')}$ and in turn, ${displaystyle (B,leq ')}$ isomorph to ${displaystyle (C,leq '')}$ then. ${displaystyle (A,leq)}$ isomorph to ${displaystyle (C,leq '')}$ .

History and concept

In the 20th century, the intuitive notion of structure has been specified in mathematics, following Aristotle's conception of matter and form, according to which each structure is a set X endowed with certain operations (such as the sum or the product) or of certain relations (such as an ordering) or certain subsets (as in the case of topology), etc. In this case, the set X is the matter and the operations, relations, etc., defined in it, are the form.

Plato's discovery that the shape is what matters is captured in mathematics with the concept of isomorphism. A map f:X→Y between two sets endowed with the same type of structure is an isomorphism when each element of Y comes from a single element of X and f transforms the operations, relations, etc., that exist in X into those that exist in And. When there is an isomorphism between two structures, both are indistinguishable, have the same properties, and any statement is simultaneously true or false. That is why in mathematics structures must be classified except for isomorphisms.

In the XX century, the Austrian biologist and philosopher of science, Ludwig von Bertalanffy, recovered this concept as an element in the formulation of his General Theory of Systems. For this author there were a series of coincidences in the evolution of the processes that are carried out in different fields of knowledge (biology, demography, physics, society, etc.) which he called isomorphism. It was important for the approach of the new theory, because "the isomorphism found between different fields is based on the existence of general principles of systems, of a more or less well-developed general theory of systems".

Partial isomorphism

It is defined by:

A partial isomorphism between two sets sorted ${displaystyle (P,leq)}$ and ${displaystyle (Q,leq ')}$ It's a bi-ective function. ${displaystyle {begin{array}{rrcl}h:Xto Q\end{array}}}$ with ${displaystyle Xsubseteq P}$ for everything ${displaystyle p_{1},p_{2}in X}$ you have to: ${displaystyle p_{1}leq p_{2}}$ Yes and only if ${displaystyle h(p_{1})leq 'h(p_{2})}$ .

Examples of Isomorphisms

For example, if X is the set of positive real numbers with the product and Y is the set of real numbers with the sum, the logarithmic function ln:X→And is an isomorphism, because ${displaystyle ln(ab)=ln(a)+ln(b)}$ and every real number is the logarithm of a single positive real number. This means that each statement on the product of positive real numbers has (without replacing each number with its logarithm) an equivalent statement in terms of the sum of real numbers, which is usually simpler.

Another example: if in the space E we choose a unit of length and three mutually perpendicular axes that meet at a point, then to each point in space we can associate its three Cartesian coordinates, thus obtaining an map f:E→R³ in the set of sequences of three real numbers. When in E we consider the distance that defines the unit of fixed length and in R³ we consider the distance that defines the square root of the sum of the squares of the differences, f is an isomorphism. This fundamental discovery of Descartes allows any problem of the geometry of space to be stated in terms of sequences of three real numbers, and this method of approaching geometric problems is the core of so-called analytic geometry.^{[citation required]}

Characteristics of isomorphism

The discovery of an isomorphism between two structures essentially means that the study of each can be reduced to that of the other, which gives us two different points of view on each question and is usually essential in its proper understanding. It also means an analogy as a form of logical inference based on the assumption that two things are the same in some respects, those on which the comparison is made. In social sciences, an isomorphism consists of the application of an analogous law because there is no specific one or also the comparison of a biological system with a social system, when it comes to defining the word "system". It is equally so the imitation or copy of a tribal structure in a habitat with an urban structure.

Morphisms

Isomorphisms of a structure with itself in a bijective manner are called automorphisms.

In general, in an arbitrary category, isomorphisms are defined as being the morphisms f:X→Y that admit an inverse map h:Y→X, both left and right inverse. They may not be bijective morphisms, as is already the case for topological spaces.

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