# Homomorphism, Injective function, Bijection and Algebraic structure

In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

- Bijection

Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomorphisms).

- Algebraic structure

A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures.

- Homomorphism

The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

- Injective function

An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.

- Homomorphism

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures.

- Injective function

For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.

- Homomorphism

Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.

- Algebraic structure

For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.

- Bijection

500 related topics

## Set (mathematics)

Mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

injective (or one-to-one) if it maps any two different elements of A to different elements of B,

bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.

For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

## Module (mathematics)

Generalization of the notion of vector space, wherein the field of scalars is replaced by a ring.

If R is a ring, we can define the opposite ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.

This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects.

A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic.

A representation is called faithful if and only if the map R → EndZ(M) is injective.

## Category theory

General theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of 20th century in their foundational work on algebraic topology.

Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure (homomorphisms).

Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of mathematical structure and studies also the relationships between structures of different nature.

## Field (mathematics)

Set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

. All field homomorphisms are injective.

(i.e., the bijections

## Inverse element

Inverse element generalises the concepts of opposite (

More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.

An invertible homomorphism is called an isomorphism.

In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions.

A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.

## Monoid

Set equipped with an associative binary operation and an identity element.

Such algebraic structures occur in several branches of mathematics.

A homomorphism between two monoids

A bijective monoid homomorphism is called a monoid isomorphism.

## Mathematics

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow.

## Vector space

The one-to-one correspondence between vectors and their coordinates vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication.

is both one-to-one (injective) and onto (surjective).

## Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

Any bijective ring homomorphism is a ring isomorphism.

## Category of sets

Category whose objects are sets.

The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

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