Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos 'equal', and μορφή morphe 'form' or 'shape') is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos 'equal', and μορφή morphe 'form' or 'shape') is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general. Isomorphisms are formalized using category theory. A morphism f : X → Y in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : Y → X in that category such that gf = 1X and fg = 1Y, where 1X and 1Y are the identity morphisms of X and Y respectively. Let R + {displaystyle mathbb {R} ^{+}} be the multiplicative group of positive real numbers, and let R {displaystyle mathbb {R} } be the additive group of real numbers. The logarithm function log : R + → R {displaystyle log colon mathbb {R} ^{+} o mathbb {R} } satisfies log ⁡ ( x y ) = log ⁡ x + log ⁡ y {displaystyle log(xy)=log x+log y} for all x , y ∈ R + {displaystyle x,yin mathbb {R} ^{+}} , so it is a group homomorphism. The exponential function exp : R → R + {displaystyle exp colon mathbb {R} o mathbb {R} ^{+}} satisfies exp ⁡ ( x + y ) = ( exp ⁡ x ) ( exp ⁡ y ) {displaystyle exp(x+y)=(exp x)(exp y)} for all x , y ∈ R {displaystyle x,yin mathbb {R} } , so it too is a homomorphism. The identities log ⁡ exp ⁡ x = x {displaystyle log exp x=x} and exp ⁡ log ⁡ y = y {displaystyle exp log y=y} show that log {displaystyle log } and exp {displaystyle exp } are inverses of each other. Since log {displaystyle log } is a homomorphism that has an inverse that is also a homomorphism, log {displaystyle log } is an isomorphism of groups. Because log {displaystyle log } is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.